The document provides an overview of microwave engineering and rectangular waveguides. It defines microwave frequencies as ranging from 1 GHz to 300 GHz. Rectangular waveguides transmit electromagnetic waves through successive reflections from inner walls. Modes in waveguides include transverse electric (TE) and transverse magnetic (TM) modes. The document analyzes the TM and TE modes in rectangular waveguides through solving Maxwell's equations with boundary conditions. Cut-off frequencies above which modes can propagate are determined. Examples demonstrate calculating waveguide parameters and resonant frequencies of cavity resonators.

1. MicrowaveMicrowave EngineeringEngineering

2. OutlineOutline
 Introduction to Microwaves
 Microwave Frequency Bands
 Advantages of Microwaves
 Applications of Microwaves
 Comparison of Transmission Line and Waveguide
 Rectangular Waveguide
 Analysis of TM modes for Rectangular Waveguide
 Analysis of TE modes for Rectangular Waveguide
 Waveguide Parameters
 Rectangular waveguide Cavity Resonators.
 Examples

3. Electromagnetic SpectrumElectromagnetic Spectrum

4. Microwave PropertiesMicrowave Properties
Microwaves frequency range 1 GHz – 300 GHz
Microwave is an electromagnetic radiation of
short wavelength.
 Reflected by conducting surfaces
Microwave currents flow through a thin outer
layer of an ordinary cable.
They are not reflected by ionosphere

5. Microwave Bands DesignationMicrowave Bands Designation
Band Frequency
(GHz)
Wavelength(cm)
L 1 to 2 30.0 to 15.0
S 2 to 4 15 to 7.5
C 4 to 8 7.5 to 3.8
X 8 to 12 3.8 to 2.5
Ku 12 to 18 2.5 to 1.7
K 18 to 27 1.7 to 1.1
Ka 27 to 40 1.1 to 0.75
Millimeter 40-300 0.75-0.1

6. Advantages of Microwaves
Increased Bandwidth Availability
Improved Directive properties
Less fading effect and more reliable
Lower power requirement
Transparency Property of Microwaves

7. Applications
 Telecommunication:
Intercontinental Telephone and TV
Space communication
Telemetry
 Radars:
Detect aircraft
Track / guide supersonic missiles
Observe and track weather patterns
Air Traffic Control (ATC)

8. Applications
 Commercial Applications
Microwave ovens
Drying machines
 Rubber / Plastics / Chemical /Food process
industry
 Biomedical Applications
 Electronic Warfare and Counter Warfare

9. Waveguides
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.

10. WaveguidesWaveguides
 Waveguides, like transmission lines, are structures
used to guide electromagnetic waves from point to
point.
 However, the fundamental characteristics of waveguide
and transmission line waves (modes) are quite different.
 The differences in these modes result from the basic
differences in geometry for a transmission line and a
waveguide.
 Waveguides can be generally classified as either metal
waveguides or dielectric waveguides.
 Metal waveguides normally take the form of an
enclosed conducting metal pipe. The waves
propagating inside the metal waveguide may be
characterized by reflections from the conducting walls.

11. Transmission line and Wave guide -1Transmission line and Wave guide -1
Transmission line Wave guide
Two or more conductors
separated by some
insulating medium (two-
wire, coaxial, micro strip,
etc.)
Metal waveguides are
typically one enclosed
conductor filled with an
insulating medium.
Normal operating mode is
the TEM or quasi-TEM
mode (can support TE and
TM modes but these modes
are typically undesirable).
Operating modes are TE or
TM modes (cannot support
a TEM mode).

12. Transmission line and Wave guide -2Transmission line and Wave guide -2
Transmission line Wave guide
No cutoff frequency for the
TEM mode. Transmission
lines can transmit signals
from DC up to high
frequency.
Must operate the waveguide
at a frequency above the
respective TE or TM mode
cutoff frequency for that
mode to propagate.
Significant signal
attenuation at high
frequencies due to conductor
and dielectric losses.
Lower signal attenuation at
high frequencies than
transmission lines.

13. Transmission line and Wave guide -3Transmission line and Wave guide -3
Transmission line Wave guide
Small cross-section
transmission lines (like
coaxial cables) can only
transmit low power levels
due to the relatively high
fields concentrated at
specific locations within the
device
Metal waveguides can
transmit high power levels.
The fields of the propagating
wave are spread more
uniformly over a larger
cross-sectional area than the
small cross-section
transmission line.
Large cross-section
transmission lines can
transmit high power levels.
Large cross-section (low
frequency) waveguides are
impractical due to large
size and high cost.

14. Rectangular WaveguideRectangular Waveguide
The walls of the waveguide are made of ideal
conductor and the medium filling the waveguide is
ideal dielectric.

15. AssumptionsAssumptions
The waveguide is infinitely long, oriented
along the z-axis, and uniform along its length.
The waveguide is constructed from ideal
materials. [perfectly conducting pipe (PEC) is
filled with a perfect insulator (lossless
dielectric)].
 Fields are time-harmonic.

16. General Wave Characteristics as DefinedGeneral Wave Characteristics as Defined
by Maxwell’s Equationsby Maxwell’s Equations
 Given any time-harmonic source of electromagnetic
radiation, the phasor electric and magnetic fields
associated with the electromagnetic waves that
propagate away from the source through a medium
characterized by (μ,ε) must satisfy the source-free
Maxwell’s equations given by

17. General Wave Characteristics as DefinedGeneral Wave Characteristics as Defined
by Maxwell’s Equationsby Maxwell’s Equations
 The source-free Maxwell’s equations can be
manipulated into wave equations for the electric and
magnetic fields. These wave equations are
where the wave number k is real-valued for lossless
media and complex valued for lossy media.

18. Types of modes in WaveguideTypes of modes in Waveguide
Transverse Magnetic (TM) wave: Here only
magnetic field is transverse to the direction of
propagation and the electric field is not purely
transverse.
(i.e.) Ez ≠ 0, Hz = 0.
Transverse Electric (TE) wave: Here only the
electric field is purely transverse to the direction
of propagation and the magnetic field is not
purely transverse.
(i.e.) Ez = 0, Hz ≠ 0.

19. Analysis of TM modesAnalysis of TM modes
 The wave equation is to be solved for Ez with
appropriate boundary conditions. In Cartesian co-
ordinates the wave equation for Ez can be written as:
 The equation can be solved by the separation of
variables i.e. by assuming that Ezis given as:
 The fields are assumed to be of sinusoidal nature with
an angular frequency ω.

20. Analysis of TM modesAnalysis of TM modes
 Substituting this solution of Ez and re-arranging the
terms, the wave equation becomes-
The first term is a function of x only, the second term
is a function of y only, the third term is a function of
z only and fourth term is a constant.
 Since the equation is to be satisfied for every value of x,
y, z in each term in equation must be constant.

21. where, A, B and β are real constants.
The parameter β is commonly referred as modal
phase constant.
Analysis of TM mode

22. Analysis of TM modesAnalysis of TM modes
 From the physical understanding of reflection of
waves from parallel conducting boundaries we expect a
standing wave kind of behavior in x and y directions
and a travelling wave kind of behavior in z direction.
 In any case, we expect a wave phenomenon in x,y,z
direction which can be properly represented by putting a
negative sign in front of the constants A2
, B2
and β2
.
 Instead of negative sign if the positive sign was used the
solutions will have real exponential functions which
would not represent the wave phenomenon.

23. Analysis of TM mode
These equations can be re-written as:
These equations are identical to the transmission
line equations.

24. Analysis of TM mode
 The solution to these equations can be appropriately
written as:
 If we assume that waveguide is of infinite length, we
can take only one travelling wave in +z-direction.

25. Analysis of TM mode
Boundary condition 1:
Ez= 0 @ x = 0 leads to C1 = 0
Boundary condition 2:
Ez= 0 @ y = 0 leads to C3 = 0
The equation of Ez now reduces to
Ez = C * sin(Ax) * sin(By) * e-jβz

26. Analysis of TM mode
The equation of Ez
Ez = C * sin(Ax) * sin(By) * e-jβz
Boundary condition 3:
Ez = 0 @ x = a leads to
Boundary condition 4:
Ez = 0 @ y = b leads to
where, m & n are integers.

27. Analysis of TM mode
Substituting for A and B, we get the solution for
Ez as:
One can get, the values of transverse
components Ex, Ey, Hx, Hy from longitudinal
components Ez and Hz.
 In case of TM mode, as Hz = 0, the all 4
transverse components can be obtained from Ez
only.

28. Analysis of TM mode
where, h2
is transverse propagation constant given by-

29. Observations for TMmn Modes
 The fields existing in the discrete electric and
magnetic field pattern called modes of waveguide.
 All field components vary sinusoidal in x and y
directions.
 All transverse fields go to zero if either m or n is
zero.
 Both the indices m and n have to be non-zero for
existence of the TM mode.
 TMm0 and TM0n modes can not exist.
 Consequently, the lowest order mode which can exist
is mode TM11 mode.

30. Analysis of TM mode
We started with ,

31. Analysis of TM mode
Re-writing this expression,
-A2
- B2
- β2
+ ω2
µε = 0
Also, we have,
which leads to,

32. Analysis of TE ModeAnalysis of TE Mode
For TE mode, Ez = 0 and only Hz is present, all
4 transverse components of the field can be
obtained from Hz only.
 In the case of TM mode, the wave equation was
solved for Ez which was tangential to all the four
walls of the waveguides. We therefore had
boundary conditions on Ez.
 In the TE case however the independent component
Hz is tangential to the walls of the waveguide which
do not impose any boundary conditions on Hz.

33. Analysis of TE Mode
 One can note that for x=0 and x=a , (vertical walls) and
for y= 0 and y=b, (horizontal walls) the tangential
component of magnetic field is maximum.
 Proceeding the analysis for TE mode similar to TM
mode, with knowledge that Hz is maximum at the
boundaries, one can obtain the solution for Hz as:

34. Analysis of TE Mode
Where, h2
is transverse propagation constant given by-

35. Analysis of TE Mode
 The fields for the TE modes have similar behavior to
the fields of the TM modes i.e. they exist in the form of
discrete pattern.
 They have sinusoidal variations in x and y
directions, indices m and n represent number of half
cycles of the field amplitudes in x and y direction
respectively.
 Unlike TM mode both indices m and n need not be
non-zero for the existence of the TE mode.
 TE00 mode cannot exist but TEm0 and TE0n modes can
exist.
 The lowest order mode for the TE case therefore is TE10
and TE01 .

36. Phase constant of TE and TM modePhase constant of TE and TM mode
For both TEmn and TMmn modes the modal phase
constant β is given by:
For the mode to be travelling β has to be a real
quantity.
If β becomes imaginary then the fields no more
remain travelling but become exponentially
decaying.

37. Cut-off Frequency of TE and TMCut-off Frequency of TE and TM
modemode
 The frequency at which β changes from real to
imaginary is called the cut-off frequency of the
mode. At cut-off frequency, therefore β = 0
gives,

38. Cut-off Frequency of TE and TMCut-off Frequency of TE and TM
modemode
 The cut-off frequencies for lowest TM and TE
modes i.e. TM11, TE10and TE01 can be obtained as:

39. Cut-off Frequency of TE and TMCut-off Frequency of TE and TM
modemode
Since by definition we have a > b we get the
frequencies as-
We can make an important observation that, if at
all the electromagnetic energy travels on a
rectangular waveguide its frequency has to be
more than the lowest cut-off frequency i.e. fc
of TE10 mode.
As the order of the mode increases the cut-off
frequency also increases.

40. Cut-offCut-off wavelength ofof
TE and TM modeTE and TM mode
 The very first mode that propagates on the rectangular
waveguide is TE10 mode and therefore this mode is called
the dominant mode of the rectangular waveguide.
 The cut-off wavelength is given by
For dominant mode , λc=2a.
For propagation of wave in the waveguide
λ < λc or f > fc
22
2






+





=
b
n
a
m
cλ

41. Field PatternField Pattern
 The visualization of the modal fields is important for
identifying regions from where fields can be tapped
efficiently by the probes.
 The field probes are devices which can induce fields
inside a waveguide or extract energy from the fields
propagating inside the waveguide.
 One can see from the modal field expression that the
fields are periodic over one guided wavelength λg along
the length of the waveguide.
 So essentially one has to develop a three dimensional
picture of the fields only over a block of λg.

43. Waveguide ParametersWaveguide Parameters
Guide Wavelength
It is defined as the distance
travelled by the wave in
order to undergo a phase
shift of 2π radians.
It is related to phase
constant by the relation
λg = 2π / β
Wave Impedance
It is defined as ratio of
strength of electric field in
one transverse direction to
the strength of magnetic
field along other transverse
direction.

44. Waveguide ParametersWaveguide Parameters
Phase Velocity
 The phase velocity is
defined as the velocity with
which the wave changes
phase in terms of the guide
wavelength
 Vp = λg * f
Group Velocity
 The group velocity of
a wave is defined as the
rate at which the wave
propagates through the
waveguide.
 Vg = dω / dβ
The product of phase and group velocities is equal to
square of the velocity of light. i.e. vp * vg = c2

45. Example 1Example 1

46. Example 2Example 2

48. Example 3Example 3

51. Example 4Example 4
 Find the resonant frequencies of the dominant mode of
an air filled rectangular cavity of dimensions 5cm x 4cm
x 2.5cm.The dominant mode for rectangular cavity
resonator is TE101
 Given a = 5 cm, b = 4 cm, d = 2.5 cm
Resonant frequency of TE and TM modes is given by
fr = 6.71 GHz

52. Thank YouThank You