GATE ACADEMY ® 157 Waveguides

3.1 Introduction
Waveguides are hollow metallic structures used for transmitting electromagnetic energy. Wave guides
and transmission lines are two common methods of transmitting EM energy from source to load in a
bounded (guided) medium.
However, waveguides differ from transmission lines in some aspects, such as
(a) A transmission line can support only a TEM wave, where as a wave guide can support many possible
field configurations (TE and TM ).
(b) At microwave frequencies (3-300 GHz), transmission lines become inefficient as a result of skin
effect and dielectric losses, whereas waveguides are used at that range of frequencies to obtain larger
bandwidth and lower signal attenuation.
(c) A transmission line may operate from dc (f = 0) to a very high frequency, whereas waveguides can
operate above a certain frequency called cut-off frequency and therefore act as high pass filter.
Waveguides cannot transmit dc, and they become excessively large at frequencies below microwave
frequencies.
(d) The whole body of wave guide acts as a ground, and wave propagates through multiple reflections.
3.2 Comparison Between Waveguide and Twin Wire Transmission Line
1. In transmission lines, cutoff frequency is zero, so it allows all the frequency signals to pass through
it, while in waveguides, only that frequency is allowed to pass, that is greater than a particular
frequency called the cutoff frequency of waveguide.
2. Wave propagation in wave guide is based on field theory, because variables in case of wave guide
are electric and magnetic field, while propagation of wave in transmission lines is based upon
electric circuit theory because variables in case of transmission lines are voltage and current.
3. In wave guides, we define wave impedance, also known as intrinsic impedance which is analogous
to characteristic impedance of transmission line.
3.3 Modes of Wave Propagation
For time-harmonic fields, assuming wave propagation along the z-axis, the electric and magnetic fields
can be written as
E( x, y, z)  [ Et ( x, y)  Ez az ] e jz
Electromagnetic Theory 158 GATE ACADEMY ®

H ( x, y, z)  [ Ht ( x, y)  H z az ] e jz

where the first terms Et ( x, y) and Ht ( x, y) represent the transverse components and the second terms
Ez and H z represent the longitudinal components of the electric and magnetic fields, respectively.
Considering the expression of field components, we define the following modes of wave propagation :
(a) Transverse Electromagnetic (TEM) Modes
In TEM mode, the electric and magnetic fields are transverse to the direction of wave propagation
with no longitudinal components, i.e.
Ez  H z  0

E H
Ex Hx
Ey Hy
Ez =0 Hz  0

(b) Transverse Electric (TE) Modes

In TE mode, the electric field is transverse to the direction of propagation (no longitudinal electric field
component), while the magnetic field has both transverse and longitudinal components, i.e.
Ez  0, H z  0

TE mode is represented as TEmn mode

where m  number of half sinusoidal cycle variations along x-axis.
n  number of half sinusoidal cycle variations along y-axis.
GATE ACADEMY ® 159 Waveguides

m, n are also referred to as Eigen values in waveguide.

E H
Ex Hx
Ey Hy
Ez  0 Hz

(c) Transverse Magnetic (TM) Modes

In TM mode, the magnetic field is transverse to the direction of propagation (no longitudinal
magnetic field component), while the electric field has both transverse and longitudinal components,
i.e. H z  0, Ez  0

E H
Ex Hx
Ey Hy
Electromagnetic Theory 160 GATE ACADEMY ®

Ez Hz  0
(d) Hybrid (HE) Modes
In this case, neither the E nor the H field is transverse to the direction of wave propagation.
Sometimes these modes are referred to as hybrid modes.
H z  0, Ez  0

Remember
1. Transverse electric (TE) and Transverse magnetic (TM) modes are commonly
referred to as waveguide modes since they are the only modes which can exist in
an enclosed guiding structure.
2. Transverse electromagnetic (TEM) modes cannot exist on single conductor guiding
structures. TE and TM modes are characterized by a cut-off frequency below
which they do not propagate.
3. TE and TM modes can exist on transmission lines but are generally undesirable
(higher order modes).
4. TEM modes are sometimes, called transmission line modes since they are the
dominant modes on transmission lines.
5. Transmission lines are typically operated at frequencies below the cut-off
frequencies of TE and TM modes so that only the TEM mode exists.
6. Quasi-TEM modes are modes which approximate true TEM modes when the
frequency is sufficiently small.
7. The mode having lowest cut-off frequency is called dominant mode.

3.4 Rectangular Waveguide

We assume that the wave guide is filled with a source free  v  0, J = 0 lossless dielectric material
 0  and that its walls are perfectly conducting  c  .

 a  b (standard rectangular waveguide)

 a  b (square waveguide)
GATE ACADEMY ® 161 Waveguides

a
  aspect ratio
b
3.5 Rectangular Waveguide Parameters
Propagation constant :
In a rectangular waveguide, the propagation constant is defined as
  h2  k 2 …. (i)
Where, k   
  h 2  2  h 2   2  2
Case 1 : Cut-off
For no propagation,   0,     0 (     j )
From equation (i), h  k 2 2

 m   n 
2 2

k 2  2      
 a  b 
At   0,   c ; c  cut-off angular frequency

 m   n 
2 2
1
c  
  a   b 
Case 2 : Evanescent
 m   n 
2 2

If k 2  2      
 a  b 
Then   ,   0
In this case, we have no wave propagation at all. These non-propagating modes are said to be
evanescent.
Case 3 : Propagation
 m   n 
2 2

If k         
2 2

 a  b 
Then   j ,   0
So, the constant  becomes

 m   n 
2 2

 k  2

 a   b 
This is the only case in which propagation takes place because all field components will have the factor
e  z  e  jz .
Cut-off frequency :
 m   n 
2 2
1
fc  
2   a   b 
The cut-off frequency is the operating frequency below which attenuations occurs and above which
propagation takes place.
Cut-off wavelength :
Electromagnetic Theory 162 GATE ACADEMY ®

2
c 
2 2
m n
   
 a  b
Below figure shows the behavior of waveguide as a High pass filter.

3.6 Rectangular Waveguide Mode Categories

TM modes ( H zs  0 ) :
For the TM case in rectangular waveguide, the non-zero field components are
j  m   mx   ny  z
Exs   2   E0 cos   sin  e
h  a   a   b 
j  n   mx   ny  z
E ys   2   E0 sin   cos  e
h  a   a   b 
 mx   ny  z
Ezs  E0 sin   sin  e
 a   b 
j  n   mx   ny  z
H xs  2   E0 sin   cos  e
h  b   a   b 
j  m   mx   ny  z  m   n 
2 2

H ys   2   E0 cos   sin  e , Where h  

2
  
h  b   a   b   a   b 
Remember
1. For TM modes in rectangular waveguides, neither m nor n can be zero.
2. TM 11 has the lowest cut-off frequency (or the longest cut-off wavelength) of all the
TM modes, therefore it is a dominant TM mode.

TE modes ( E zs  0 ):
For the TE case in rectangular waveguide, the non-zero field components are
j  n   mx   ny  z
Exs  2   H 0 cos   sin  e
h  b   a   b 
GATE ACADEMY ® 163 Waveguides

j  m   mx   ny  z

E ys   2   H 0 sin   cos  e
h  a   a   b 
j  m   mx   ny  z
H xs  2   H 0 sin   cos  e
h  a   a   b 
j  n   mx   ny  z
H ys  2   H 0 cos   sin  e
h  b   a   b 
 mx   ny  z  m   n 
2 2

H zs  H 0 cos   cos  e , Where h 2     

 a   b   a   b 
Remember
1. TE10 has the lowest cut-off frequency (or the longest cut-off wavelength) of all the
TE modes, therefore it is a dominant TE mode.
2. Two or more modes having the same cut-off frequency, are Degenerated modes. In
a rectangular waveguide the corresponding TEmn and TM mn mode are always
degenerate.

3.7 Summary of Guided Parameters in Rectangular Waveguide

Parameter TE Modes TM Modes
Dominant mode TE10 TM 11
Ez  0, H z  0 Ez  0, H z  0
Ez , H z  mx   ny   mx   ny 
H z  H 0 cos   cos   Ez  E0 sin   sin  
 a   b   a   b 
vp vp
Guided phase v pz  v pz 
2 2
   
velocity 1   1  
 c   c 
2 2
Guided group  
vgz  v p 1   vgz  v p 1  
 c   c 
velocity
 
g  g 
Guided 2 2
   
wavelength 1   1  
 c   c 
2 2
Guided phase  
g   1    g   1   
 c   c 
constant

TE  2
2 
Guided intrinsic   TM   1  
1    c 
impedance  c 
TM  
TE  
Electromagnetic Theory 164 GATE ACADEMY ®

Guided intrinsic
impedance
vs
Wavelength

Remember
1. TE TM  2
  TE TM
Where TE  guided intrinsic impedance in TE mode
TM  guided intrinsic impedance in TM mode
r
  120
r
2. Phase velocity is the velocity at which locus of the constant phase propagates
down the waveguide. It is referred to as Virtual velocity. Phase velocity can be
greater than velocity of light.
3. Group velocity is the velocity at which the resultant repeated reflective waves
propagate down the waveguide. It is velocity of propagation of the wave of the
group of frequencies. Group velocity is referred to as energy propagation velocity
in the waveguide. It is also referred to as true velocity in the waveguide.

Example 1
In a rectangular waveguide for which a  1.5cm , b  0.8cm ,   0,   0 , and   40 ,
 x   3y 
H x  2sin   cos   sin  10 t   z  A/m
11

 
a  b 
Determine :
(i) The mode of operation (ii) The cut-off frequency
(iii) The phase constant  (iv) The propagation constant 
(v) The intrinsic wave impedance 
x   3y 
Sol. Given : H x  2sin   cos   sin  10 t   z  A/m
11

 
a  b 
(i) From the given expression for H x , comparing with
 mx   ny 
H x  H 0 sin   cos   sin  t  z  A/m
 a   b 
m  1, n  3, so the guide is operating at TM13 or TE13 . Ans.
(ii) Cut-off frequency is given by,
vp m2 n 2
f cmn  
2 a 2 b2
GATE ACADEMY ® 165 Waveguides

1 vc v
vp    c
 r r 2
3 108 1 9
f c13  2
 2
4 1.5 102  0.8 102 
3 108
f c13 
4

0.444  14.06 102  28.57GHz  Ans.

(iii) Guided phase constant is given by,

2 2
f   r f 
 g    1   c   1  c 
f  c f 
100 109
  2f  1011 or f   50GHz
2
 1011  2 
2
 28.57 
 1   1718.81 rad/m Ans.
3 10  50 
8

(iv) Propagation constant is given by,

    j
For propagation,
  0,   j
   j1718.81/m Ans.
(v) Guided intrinsic impedance is given by,
2
f 
2
377  28.57 
TM13  ' 1  c   1   154.7  Ans.
f  r  50 

Example 2
The cross section of the rectangular waveguide is 20 cm  5 cm. Find 6 lowest order modes which will
propagate on the waveguide and their cut-off frequencies.
Sol. Given : For the given waveguide a  0.20 m and b  0.05 m
The cut-off frequency of a mode is given by,
1  m   n 
2 2

c     
  a   b 
2 2
 m   n 
2f c  3108    
 0.2   0.05 
2 2
 m  n
f c  1.5 10     
10

 20   5 
The cut-off frequency in ascending order will correspond to
m  1 n  0, m  4 n  0
m  2 n  0, m  0 n  1
m  3 n  0, m  1 n  1
Since, for TM modes m and n both have to be non-zero, we get the 6 lowest order modes as,
TE10 , TE20 , TE30 , TE40 , TE01 , TE11 , TM11 .
Electromagnetic Theory 166 GATE ACADEMY ®

In fact TE11 and TM 11 have same cut-off frequency. The cut-off frequencies of the modes are
Mode Cut-off Mode Cut-off
frequency frequency
TE10 0.75 GHz TE20 1.5 GHz
TE30 2.25 GHz TE40 3.0 GHz
TE01 3.0 GHz TE11 3.092 GHz
TM11 3.092 GHz Ans.
Example 3
A section of a rectangular waveguide of cross-section 2 cm  1.5 cm is to be used as a delay line in a
radar at 10 GHz. What should be the length of the section for the delay of 10 nsec ?
Sol. Given : a = 2 cm, b = 1.5 cm, f  10 GHz
Normally the waveguide is operated in the TE10 mode. The cut-off frequency of the TE10 mode is
1  3  108 
c    1.51010 rad/s
 a 0.02
The guided phase constant of the mode is given by,
  
2

g  1  c 
vc 
The group velocity of the mode is given by,
 vc
vgz  
 g  c 
2

1  

At 10 GHz,   21010 rad/sec
vc
vgz   1.512 vc  4.54 108 m/s
1   0.75 
2

For a delay of 10 nsec   108 sec  , the length of the waveguide is

L  vgz t  4.54 108 108  4.54 m Ans.

Test 1
Q. 1 The dominant mode in a waveguide is characterized by [IES EC 1993]
(A) Longest cutoff wavelength (B) Shortest frequency
(C) Infinite attenuation (D) Zero attenuation
Ans. A
Q.2 Consider the following features : [IES EC 1996]
1. Easier to use.
2. Lower power losses.
3. Higher operating frequency possible.
The advantages of waveguides over coaxial lines would include
(A) 1and 2 (B) 1 and 3 (C) 2 and 3 (D) 1, 2 and 3
Ans. C
Q.3 The lowest TM mode in a rectangular waveguide of cross-section a  b with a  b will be
GATE ACADEMY ® 167 Waveguides

[IES EC 1996]
(A) TM 01 (B) TE10 (C) TM 12 (D) TE11
Ans. C
Q.4 In a hollow rectangular waveguide, the phase velocity [IES EC 1997]
(A) Increases with increasing frequency.
(B) Decreases with increasing frequency.
(C) In independent of frequency.
(D) Will vary with frequency depending upon the frequency range.
Ans. B
Q.5 In rectangular waveguide, with a  2b, if the cut off frequency for TE20 mode is 16 GHz then the cut-off
frequency for the TM 11 mode will be [IES EC 1997]
(A) 32 GHz (B) 8 GHz (C) 4 3 GHz (D) 8 5 GHz
Ans. D
Q.6 Evanescent mode attenuation in a waveguide depends upon the [IES EC 1997]
(A) Conductivity of the dielectric filling wave guide.
(B) Operating frequency.
(C) Conductivity of the guide walls.
(D) Standing waves in the guide.
Ans. B
Q.7 For the dominant mode, in a rectangular waveguide with breadth 10 cm.The guide wavelength for a
signal of 2.5 GHz will be [IES EC 1998]
(A) 12 cm (B) 15 cm (C) 18 cm (D) 20 cm
Ans. B
Q.8 Which one of the following statements is correct? A wave guide can be considered to be analogous to a
[IES EE 2004]
(A) Low pass filter (B) High pass filter (C) Band pass filter (D) Band stop filter
Ans. B
Q.9 A standard air filled waveguide WR-187 has inside wall dimensions of a = 4.755 cm and b = 2.215 cm.
At 12 GHz, it will support [IES EC 2010]
(A) TE10 mode only (B) TE10 and TE20 modes only
(C) TE10 , TE20 and TE01 modes only (D) TE10 , TE20 , TE01 and TE11 modes
Ans. D

3.8 Power Transmission and Attenuation in Waveguide

Average poynting vector along the z-direction is given by,
1
Savg  Re[ E  H  ]
2
E2
 Savg  0 az
2
Consider TEmn , Ez  0
Electromagnetic Theory 168 GATE ACADEMY ®
2
Ex 0  E y 0
2

 Savg  az
2TE

Guided intrinsic impedance, TE 
2
 
1  
 c 
r
Where TE  0
r
Power
Savg 
Area
2
Ex 0  E y 0
a b 2

Power   
x0 y 0
2TE
dx dy

ds  dx dy az
E02
 Savg  az
2TE
E02
Savg .ds 
dx dy
2TE
1. When the dielectric medium is lossy (  0) and the waveguide walls are not perfectly conducting
(c  ) then there is continuous loss of power as wave propagates along the waveguide.
2. When EM wave propagates through a waveguide, it consists attenuation arising from one or more of
the following reasons:
(a) Losses in the dielectric (  0) .
(b) Losses in the guide walls due to finite conductivity (c  ) .
(c) Operating frequency is less than the cut-off frequency.
3. Attenuation due to cut-off frequency is given by
 
2
2
cut-off  1  c 
c   
3.9 Power Loss in Waveguide
Power flow in waveguide is given by,
Pa  P0e 2 z …. (i)
Where,   c   d
Power loss per unit length

2  Power transmitted
P
 L
2 PT
P0  Pa  PL …. (ii)
Where, PL  Power loss or power dissipated
Pa  Power delivered to antenna
GATE ACADEMY ® 169 Waveguides

P0  Input power to the waveguide

From equation (i), and (ii)
P0  PL  Pa
 P0  PL  P0e 2 z
P0 (1  e 2 z )  PL
PL  P0 (1  e 2z )
Above expression indicates Power loss in terms of input power to the waveguide.
3.10 Degenerate Modes
If two or more modes have the same cut-off frequency, then the modes are called degenerate modes.
e.g. TE11 and TM 11 are degenerate modes because these modes have same cut-off frequency.
TE12 and TM12 are degenerate modes and so on.

In case of waveguide if we take square dimension (a = b), instead of rectangular dimension (a > b), in
that case number of degenerate modes in square dimension is twice the number of degenerate mode in
rectangular dimension therefore rectangular dimension is preferred over square dimension to suppress
degenerate mode.
For example, in rectangular dimension, TE21 has TM 21 as degenerate mode (overall degenerate modes
are two).
In square dimension, TE21 has TM 21, TE12 , TM12 as degenerate modes (overall degenerate modes are
four).
Degenerate mode can be suppressed by following ways :
(i) By selecting proper dimension.

a
2
Where,   operating wavelength
a  broader wall dimension
(ii) By using mode filter.
3.11 Mode Filter
Mode filter is metallic plate which is connected across the cross-section of the waveguide. In this case
electric field will be tangential on metallic plate.
As a result that component of electric field vanishes and when electric field will disappear
corresponding magnetic field will also disappear and that mode will be filtered out from the waveguide.
Electromagnetic Theory 170 GATE ACADEMY ®

Suppress TE10 for this mode, electric field will be tangential to conductor and hence magnetic field is
zero.

Fig. TE10 suppressed Fig. TE11 suppressed

Test 2
Q.1 The degenerate modes in a waveguide are characterized by [IES EC 1998]
(A) Same cut-off frequencies and different field distributions.
(B) Same cut-off frequencies and same field distributions.
(C) Different cut-off frequency but same field distributions.
(D) Different cut-off frequency and different field distributions.
Ans. A
Q.2 When a particular mode is excited in a waveguide there appears an extra electric component in the
direction of propagation. The resulting mode is [IES EC 1993]
(A) Longitudinal electric (B) Transverse electromagnetic
(C) Transverse magnetic (D) Transverse electric
Ans. C
Q.3 A rectangular waveguide 2.29 cm 1.02 cm operates at a frequency of 11 GHz. In TE10 mode if the
maximum potential gradient of the signal is 5 kV/cm , then the maximum power handling capacity of
the wave guide will be [IES EC 1999]
(A) 31.11 mW (B) 31.11 W (C) 31.11 kW (D) 31.11 MW
Ans. C
Q. 4 The waveguide modes are usually excited from a signal source through [IES EC 2013]
(A) An antenna (B) An aperture
(C) A coaxial cable (D) Free–space coupling
Ans. C
GATE ACADEMY ® 171 Waveguides

3.12 Circular Waveguide

1. In case of circular waveguide, TE11 is dominant mode.

2. The mode representation in case of circular waveguide is TEmn are TEmn where ‘m’ denotes number
of full wave intensity variation across the circumference of cylinder and ‘n’ represents number of
half wave intensity variation which changes radially outward from center to wall of waveguide.
3. The circular waveguide is easier to manufacture as compared to Rectangular waveguide.
4. Circular waveguide occupies more space as compared to rectangular waveguide.
5. The wave impedance of a circular waveguide remains same as that of rectangular waveguide.
6. TEM mode does not exist in case of circular waveguide, because when Ez and H z will be zero,
then all the electric field and magnetic field components will also become zero.
7. The boundary condition for circular waveguide is same as rectangular waveguide which means at the
surface of conductor, only tangential magnetic field will exist, while normal component of magnetic
field will be zero and normal component of electric field will exist, while tangential component of
electric field will be zero and to satisfy this boundary conditions, we use Jacobian function.
8. At the surface of circular waveguide where r  a , Ez component should be zero and Ez is
expressed in terms of Jacobin function. So for Ez to be zero, Jacobian function should also be zero
and the root of the Jacobin function will be ( K a ) mn .
9. On the surface of circular waveguide “ E  0 ” and “ E ” component is expressed in terms of
complementary Jacobian function. For E  0 the complementary function should also be zero and
root of complementary Jacobin function will be ( K a ) mn . Cut-off wavelength for TE mode is given
by
2a
c 
( K a ) 'mn
For TE11 mode (dominate mode),
( Ka )'mn  1.84
10. Cut-off wavelength for TM mode is given by
2a
c 
( K a ) mn
Electromagnetic Theory 172 GATE ACADEMY ®

11. The average power for circular waveguide is given by,

 
2 b
1
2 0 y0
2 2
Power  E y  E  d d 

Where,  may be in TE or TM mode.

Field pattern for TE11 is given by,

Fig. Field pattern for TE11 mode

Test 3
Q.1 Figure (A), (B), (C) and (D) given below represents patterns of electric field of TE waves in circular
waveguides. Which one of them represents TE02 mode? [IES EC 1993]

(A) (B)

(C) (D)

Ans. B
Q. 2 Which one of the following modes has the characteristic of attenuation becoming less as the frequency
is increased and is attenuates at microwave frequencies of circular cylindrical waveguides?
[IES EC 1993]
(A) TE10 mode (B) TM 01 mode
(C) TE01 mode (D) higher order mode
Ans. C
Q.3 The dominant mode in a circular waveguide is [IES EC 2005]
(A) TEM mode (B) TM 01 mode (C) TE21 mode (D) TE11 mode
Ans. D
Q.4 Consider the following statements : [IES EC 2017]
Plane wave propagation through a circular waveguide result in
1. TE modes
GATE ACADEMY ® 173 Waveguides

2. TM modes
Which of the above statements is/are correct?
(A) 1 only (B) 2 only (C) Either 1 or 2 (D) both 1 and 2
Ans. C

3.18 Practice Objective Questions

Q.1 A rectangular waveguide 2.29 1.02cm operates at a frequency of 11 GHz in TE10 mode. If the
maximum potential gradient of the signal is 5 kV/cm, then the maximum power handling capacity of the
waveguide in kW will be _________.
Q.2 Phase velocity ' v p ' and the group ' v g ' in a waveguide ('c' is velocity of light) are related as
(A) v p vg  c 2 (B) v p  vg  c
(C) v p / vg  a constant (D) v p  vg  a constant
Q.3 For a hollow waveguide, the axial current must necessarily be
(A) A combination of conduction and displacement currents.
(B) Conduction current only.
(C) Time-varying conduction current and displacement current.
(D) Displacement current only.
Q.4 For a wave propagating in an air filled rectangular waveguide
(A) Guided wavelength is never less than the free space wavelength.
(B) Wave impedance is never less than the free space impedance.
(C) TEM mode is possible if the dimensions of the waveguide are properly chosen.
(D) Propagation constant is always a real quantity.
Q.5 A waveguide filled with a material whose r  2.25 has dimensions a  2 cm and b  1.4 cm. If the
guide is to transmit 10.5 GHz signals, which mode cannot be used for transmission ?
(A) TE20 (B) TE01 (C) TM11 (D) TM 13
Q.6 For TE or TM modes of propagation in bounded media, the phase velocity
(A) Is independent of frequency.
(B) Is a linear function of frequency.
(C) Is a non-linear function of frequency.
(D) Can be frequency-dependent or frequency independent depending on the source.
Q.7 A waveguide operated below cut-off frequency can be used as
(A) A phase shifter (B) An attenuator (C) An isolator (D) None of these
Q.8 At microwave frequencies, we prefer waveguides to transmission lines for transporting EM energy
because of all the following except that
(A) Losses in transmission lines are prohibitively large.
(B) Waveguides have larger bandwidths and lower signal attenuation.
(C) Transmission lines are larger than waveguides.
(D) Transmission lines support only TEM mode.
Q.9 An evanescent mode occurs when
(A) A wave is attenuated rather than propagated.
(B) The propagation constant is purely imaginary.
(C) m = 0 = n so that all field components vanish.
(D) The wave frequency is the same as the cut-off frequency.
Electromagnetic Theory 174 GATE ACADEMY ®

Q.10 For TE30 mode, which of the following field components exist ?
(A) Ex (B) H x (C) Ez (D) H y
Q.11 If in a rectangular waveguide for which a = 2b, the cut-off frequency for TE02 mode is 12 GHz, the cut-
off frequency for TM 11 mode is ____________ GHz.
Q.12 An air-filled rectangular waveguide has cross-sectional dimensions a = 6 cm and b = 3 cm.
Given that
 2x   3y 
Ez  5sin   cos 10 t  z  V/m .
12
 sin 
 a   b 
The intrinsic impedance of this mode in  is_________.
Common Data Questions 13 to 16
In air-filled rectangular waveguide, a TM mode operating at 6 GHz has
 2x   y 
Ez  5sin   sin   sin  t  12 z  V/m
 a   b 
Q.13 The mode of operation will be
(A) TM 21 (B) TE21 (C) TE12 (D) TM 12
Q.14 The cut-off frequency is _________ GHz.
Q.15 The intrinsic impedance in k is___________.
Q.16 The magnetic field H x will be
(A) 1.267sin(mx /a) cos(ny /b)sin(t  z ) mA / m
(B)  2.267sin(mx /a) cos(ny /b)sin(t  z ) mA / m
(C) 2.267sin(mx /a) cos(ny /b)sin(t  z ) mA / m
(D) 1.267sin(mx /a) cos(ny /b)sin(t  z ) mA / m
. Common Data Questions 17 to 19 .
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
 2x   3y 
E y  sin   sin 1010 t  z  V/m
10
 cos 
 a   b 
Q.17 The operating mode will be
(A) TE23 (B) TM 23 (C) TM 13 (D) TE13
Q.18 The propagation constant  will be
(A) j 200.7 / m (B) j 300.7 / m (C) j 400.7 / m (D) j 500.7 / m
Q.19 The intrinsic impedance  in ohm will be____________.
Q.20 A rectangular waveguide with cross sections shown in below figure has dielectric discontinuity. The
standing wave ratio if the guide operates at 8 GHz in the dominant mode will be _______.
GATE ACADEMY ® 175 Waveguides

Q.21 An air-filled rectangular waveguide of dimensions a = 4 cm, b = 2 cm transports energy in the dominant
mode at a rate of 2 mW. If the frequency of operation is 10 GHz, the peak value of the electric field in
the waveguide will be ________ Volt/m.
. Common Data Questions 22 to 24 .
In a rectangular waveguide for which a = 1.5 cm, b = 0.8 cm,  = 0,   0 , and   40 ,
 x   3y 
H x  2sin   cos   sin(10 t  z ) A / m
11

  
a b 
Q.22 The mode of operation is :
(A) TE13 (B) TM13
(C) Either TE13 or TM13 (D) Neither TM13 nor TE13
Q.23 The cut-off frequency is __________ GHz
Q.24 The phase constant  is ____________ rad/m
. Common Data Questions 25 to 27 .
At 15 GHz, an air-filled (5  2) cm2 waveguide has
Ezs  20sin 40x sin 50ye jz V/ m
Q.25 What mode is being propagated ?
(A) TM 21 (B) TM12 (C) TE21 (D) TE12
Q.26 The phase constant  is __________ rad/m

Q.27 The expression for E y /Ex will be

(A) 0.25 tan(40x) cot(50y ) (B) 1.25 tan(40y ) cot(50x)
(C) 0.25 tan(40y ) cot(50x) (D) 1.25 tan(40x) cot(50y )
. Common Data Questions 28 and 29 .
A standard air-filled rectangular waveguide with dimensions a = 8.636 cm, b = 4.318 cm is fed by a 4
GHz carrier from a coaxial cable.
Q.28 The phase velocity is _________ 108 m / sec .
Q.29 The group velocity is ________ 108 m / sec

Q.30 The group velocity of a 32 GHz signal propagating in the TM12 mode in an air dielectric X-band
waveguide with dimension a = 2.286 cm and b = 1.016 cm will be _________ 107 m / s .
. Common Data Questions 31 and 32 .
In a rectangular waveguide for which a = 1.5 cm, b = 0.8 cm,  = 0,   0 ,
A microwave transmitter is connected by an air-filled waveguide of cross section 2.5cm 1cm to an
antenna. For transmission at 11 GHz.
Electromagnetic Theory 176 GATE ACADEMY ®

Q.31 The ratio of the phase velocity to the medium velocity is ___________.
Q.32 The ratio of the group velocity to the medium velocity is ___________.

. Common Data Questions 33 and 39 .

A rectangular waveguide is filled with polyethylene (  2.250 ) and operates at 24 GHz. If the cut-off
frequency of a certain TE mode is 16 GHz.
Q.33 The group velocity is ___________ 108 m / s .
Q.34 The intrinsic impedance of the mode in  will be __________.
Q.35 When the electric field is at its maximum value, the magnetic energy of a cavity is :
1
(A) At its maximum value (B) At of its maximum value
2
1
(C) At of its maximum value (D) Zero
5
Q.36 A hollow rectangular waveguide has dimensions a = 4 cm, b = 2 cm. The amount of attenuation, if the
frequency is 3 GHz will be ___________dB/m.
Q.37 When the dominant H-mode is propagated in an air-filled rectangular waveguide, the guide wavelength
for a frequency of 9 GHz is 4.0 cm. The breadth of the guide will be __________ cm.
Q.38 A waveguide has an internal breadth “a” of 3 cm and carries the dominant mode of a signal of unknown
frequency. If the characteristic impedance is 500 ohm, this frequency is _________MHz.
Q.39 If a TE11 mode has to be propagated through a circular waveguide with a cut-off wavelength of 0.08
metres, the required size of the guide in cm is ___________.
. Common Data Questions 40 to 43 .
A TE wave propagating in a dielectric-filled waveguide of unknown permittivity has dimensions
a  5 cm and b  3 cm . If the x-component of its electric field is given by

Ex  36cos  40x  sin 100y  sin  2.4109 t  52.9 z  V/m

Q.40 The mode number is

(A) m  3, n  2 (B) m  2, n  3 (C) m  1, n  2 (D) m  2, n  1
Q.41 The r of material in the guide is _________.
Q.42 The cut-off frequency is _________ GHz.
Q.43 The intrinsic impedance for TE mode is ___________  .
. Common Data Questions 44 to 50 .
An air filled rectangular waveguide ( a  2.286 cm, b  1.016 cm ) provides single mode operation.
Q.44 The frequency range over which waveguide provides single mode operation is
(A) 1.26 GHz to 5.29 GHz (B) 6.56 GHz to 13.12 GHz
(C) 14.32 GHz to 20.03 GHz (D) 21.89 GHz to 25.44 GHz
Q.45 If this waveguide is filled with a nonmagnetic dielectric in order to decreases the cut-off frequency of
the dominant mode to 70% of its original value, the relative permittivity of the dielectric will be ______.
GATE ACADEMY ® 177 Waveguides

Q.46 For the air-filled waveguide, the propagation constant  mn for the TE10 , mode when the waveguide is
operated at f  0.9 f c10 should be ________ m 1 .
Q.47 For the air filled waveguide operating at 10 GHz, the waveguide wavelength is _______ cm
Q.48 For the air filled waveguide operating at 10 GHz, the phase velocity is ________ 108 m/s
Q.49 For the air filled waveguide operating at 10 GHz, the group velocity is ________ 108 m/s
Q.50 For the air filled waveguide operating at 10 GHz, the wave impedance in  is _______.
. Common Data Questions 51 to 53 .

A brass waveguide (c  1.1107 S/m) of dimensions a  4.2 cm , b  1.5cm is filled with Teflon
( r  2.6,   1015 S/m) . The operating frequency is 9 GHz. For the TE10 mode :

Q.51 The dielectric loss (  d ) in Nep/m is _________ 1013 .

Q.52 The conductor loss (  c ) in Nep/m is ___________ 10 2 .

Q.53 The loss in decibels in the guide if it is 40 cm long is _________ .

Q.54 For TM waves in a parallel plate waveguide, the minimum attenuation arising from imperfect conductors
would occur at a frequency of ( f c is the cut-off frequency) [IES EC 1998]
(A) 3 fc (B) 3 fc (C) 2 f c (D) 2 fc
Q.55 How is the attenuation factor in parallel plate guides represented ? [IES EE 2007]
(A)  = power lost / power transmitted
(B)  = 2  power lost / power transmitted
(C)  = power lost per units length / ( 2  power transmitted)
(D)  = power lost / (power lost + power transmitted)
Q.56 In a parallel plate waveguide, what is the principal wave? [IES EC 2008]
(A) TEM wave (B) TE wave
(C) TM wave
(D) Combination of TE and TM waves having axial components of both electric and magnetic fields
Q.57 For parallel plane waveguides, which is the mode with lowest cut-off frequency?
[IES EC 2009]
(A) TE10 (B) TM 10 (C) TEM (D) TM 10
Q.58 A signal propagated in a waveguide has a full wave of electric intensity change between the two walls
and no component of the electric field in the direction of propagation. The mode is
[IES EC 1993]
(A) TE11 (B) TE10 (C) TM 22 (D) TE20
Q.59 Figures 1, 2, 3, 4 show transverse electric field lines. For TE modes in a rectangular wave guide:
Electromagnetic Theory 178 GATE ACADEMY ®

Match the TE modes with the figures and select the correct answer using the codes given below
[IES EC 1994]
(A) TE10 TE02 TE12 TE21 (B) TE11 TE20 TE21 TE22
(C) TE10 TE20 TE11 TE21 (D) TE10 TE10 TE10 TE11
Q.60 The field configuration in two different views of waves propagating in a rectangular waveguide as
shown in the given figure represents [IES EC 1995, 1996]

(A) TE10 mode (B) TM 11 mode (C) TE11 mode (D) TE21 mode
Q.61 If v is the velocity of propagation in an unbounded medium, v p , and vg are the phase and group velocities
in a guide filled with a medium, having the same permittivity as that of the unbounded medium, then
v, v p ,and vg are related as [IES EC 1997]
(A) v pv  vg2 (B) vg v  v 2p (C) v p vg  v2 (D) (v  v p )(v  vg )  v2
Q.62 A dominant mode waveguide, not terminated in its characteristic impedance, is excited with a 10 GHz
signal. If ‘d’ is the distance between two successive minima of the standing wave in the guide, then
[IES EC 1997]
(A) d  1.5 cm (B) d is less than 1.5 cm
(C) d is greater than 1.5 cm (D) d  3 cm
Q.63 For a rectangular waveguide (a  b, a  b) to support only the TE10 mode at wave length  . Which one
of the following pairs of inequalities is to be satisfied? [IES EC 1998]
GATE ACADEMY ® 179 Waveguides

(A) b    2b,   2a (B) b    2b,   2a

(C) a    2a,   2b (D) a    2a,   2b
Q.64 The cut-off wavelength  c for TE20 mode for a standard rectangular waveguide is
[IES EC 2000]
2
(A) (B) 2a (C) a (D) 2a 2
a
Q.65 For a wave propagating in an air filled rectangular waveguide [IES EC 2002]
(A) Guided wavelength is never less than free space wavelength.
(B) Wave impedance is never less than the free space impedance.
(C) TEM mode is possible if the dimensions of the waveguide are properly chosen.
(D) Propagation constant is always a real quantity.
Q.66 Which one of the following statements is correct? The wavelength of a wave propagating in a wave
guide is [IES EE 2004]
(A) Smaller than the free space wavelength. (B) Greater than the free space wavelength.
(C) Directly proportional to the group velocity. (D) Inversely proportional to the phase velocity.
Q.67 The cut-off frequency of the dominant mode of a rectangular waveguide having aspect ratio more than 2
is 10 GHz. The inner broad wall dimension is given by [IES EC 2005]
(A) 3 cm (B) 2 cm (C) 1.5 cm (D) 2.5 cm
Q.68 In a waveguide, the evanescent modes are said to occur if [IES EC 2005]
(A) The propagation constant is real. (B) The propagation constant is imaginary.
(C) Only the TEM waves propagate. (D) The signal has a constant frequency.
Q.69 Which one of the following modes has the highest cut-off wavelength in a rectangular waveguide?
[IES EC 2006]
(A) TE10 (B) TE01 (C) TM 01 (D) TM 11
Q.70 Which one of the following modes has the highest cut-off wavelength in a rectangular waveguide ?
[IES EC 2006]
(A) TE10 (B) TE01 (C) TM 01 (D) TM11
Q.71 If  is the width of the rectangular wave guide and  is the wavelength, then which one of the
following is correct ? [IES EE 2007]
(A)    / 4 but   / 2 (B)    / 2 but  3 / 2
(C)   3 / 2 (D)    / 4
Q.72 Inside a waveguide with perfectly conducting walls, any current present is in the form of
(A) Displacement current only. [IES EC 2008]
(B) Conduction current only.
(C) Partially displacement current and partially conduction current.
(D) Sometimes displacement current and sometimes conduction current.
Q.73 In a rectangular waveguide [IES EC 2008]
(A) TE and TEM waves can exist but TM waves cannot exist.
(B) TM and TEM waves can exist but TE waves cannot exist.
(C) TE and TM waves can exist but TEM waves cannot exist.
(D) TE, TM and TEM all can exist.
Q.74 A standard waveguide WR90 has inside wall dimensions of a = 2.286 cm and b = 1.016 cm. What is the
cut-off wavelength for TE01 mode? [IES EC 2009]
Electromagnetic Theory 180 GATE ACADEMY ®

(A) 4.572 cm (B) 2.286 cm (C) 2.032 cm (D) 1.857 cm

Q.75 The dominant mode in rectangular waveguides is [IES EC 2009]
(A) TE10 (B) TE11 (C) TM 01 (D) TM 11
Q.76 Consider the following statements [IES EC 2009]
For a square waveguide of cross-section 3cm  3cm it has been found
1. At 6 GHz dominant mode will propagate.
2. At 4 GHz all the modes are evanescent.
3. At 11 GHz only dominant modes and no higher order mode will propagate.
4. At 7 GHz degenerate modes will propagate.
Which of the above statements are correct ?
(A) 1 and 2 only (B) 1, 2 and 3 only (C) 2 and 3 only (D) 2, 3 and 4
Q.77 Match List-I with List-II and select the correct answer using the code given below the lists :
[IES EC 2009]
List-I List-II
(Modes) (Characteristic)
A. Evanescent mode 1. Rectangular waveguide does not support
B. Dominant mode 2. No wave propagation
C. TM10 and TM 01 3. Lowest cut-off frequency
Codes :
A B C
(A) 1 2 3
(B) 2 3 1
(C) 1 3 2
(D) 2 1 3
Q.78 Assertion (A) : A z-directed rectangular waveguide with cross-sectional dimensions 3 cm  1 cm will
support propagation at 4 GHz.
 m   n   2 
2 2 2

Reason (R) : K  
2
     
 3   1    
z

where  is the wavelength. [IES EC 2009]

Codes :
(A) Both A and R are individually true and R is the correct explanation of A.
(B) Both A and R are individually true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.
Q.79 The mode with lowest cut-off frequency for an electromagnetic wave propagation between two perfectly
conducting parallel plates of infinite extent is [IES EC 2010]
(A) TE10 (B) TM10 (C) TM 01 (D) TEM
Q.80 Consider the following statements [IES EC 2010]
For a rectangular waveguide with dimensions a  b where b is the narrow dimension, small value of b
1. Gives a larger separation between cut-off frequencies of TE01 and TE10 modes.
GATE ACADEMY ® 181 Waveguides

2. Gives increased attenuation.

3. Limits power handling capabilities because of breakdown field limits.
Which of the above statements is/are correct?
(A) 1 and 2 only (B) 1, 2 and 3 (C) 2 only (D) 3 only
Q.81 An air-filled rectangular waveguide has dimensions of a = 6 cm and b = 4 cm. The signal frequency is 3
GHz. Match List-I with List-II and select the correct answer using the code given below the lists :
[IES EC 2010]
List-I List-II
A. TE10 1. 2.5 GHz
B. TE01 2. 3.75 GHz
C. TE11 3. 4.506 GHz
TM 11
D. 4. 4.506 GHz
Codes :
A B C D
(A) 1 2 3 4
(B) 4 2 3 1
(C) 1 3 2 4
(D) 4 3 2 1
Q.82 Consider a rectangular waveguide of internal dimensions 8cm  4 cm . Assuming a H10 mode of
propagation, the critical wavelength would be [IES EC 2012]
(A) 8 cm (B) 16 cm (C) 4 cm (D) 32 cm
 m   n 
2 2

Q.83           represent propagation constant in a rectangular waveguide for

2

 a   b 
[IES EC 2012]
(A) TE waves only (B) TM waves only
(C) TEM waves (D) TE and TM waves
Q.84 The cut-off frequency of a rectangular waveguide in dominant mode is 10 GHz. The width of the wave
guide is [IES EC 2015]
(A) 2 cm (B) 1.5 cm (C) 1 cm (D) 2.5 cm

3.19 Practice Objective Answer Keys

1. 31.11 2. A 3. D 4. A 5. D 6. C 7. B 8. C

9. A 10. B 11. 6.7 12. 375 13. A 14. 5.973 15. 3.978 16. D

17. A 18. C 19. 985.3 20. 1.65 21. 63.77 22. C 23. 28.57 24. 1718.81

25. A 26. 241.3 27. D 28. 3.33 29. 2.7 30. 9.791 31. 1.193 32. 0.831

33. 1.491 34. 337.2 35. D 36. 409.11 37. 2.26 38. 7614 39. 2.345 40. B
Electromagnetic Theory 182 GATE ACADEMY ®

41. 2.25 42. 2.4 43. 569.9 44. B 45. 2.04 46. – 59.9 47. 3.97 48. 3.97

49. 2.26 50. 499 51. 1.205 52. 1.744 53. 0.0606 54. D 55. C 56. A

57. C 58. D 59. C 60. C 61. C 62. C 63. D 64. C

65. A 66. B 67. C 68. A 69. A 70. A 71. A 72. A

73. C 74. C 75. A 76. A 77. B 78. D 79. D 80. B

81. A 82. B 83. D 84. B

GATE ACADEMY ® 183 Waveguides

3.20 GATE Conventional Questions

Q.1 A rectangular waveguide with cross-section dimensions 5 cm by 3 cm is filled with a dielectric of
relative permittivity 3. [GATE EC 1988 : 8 Marks]
(i) Determine the cutoff frequency of the TE11 mode.
(ii) Determine the frequency at which this mode has an attenuation of 3 Np/m.
Q.2 A rectangular hollow metal waveguide of internal cross section of 7.366 cm  3.556 cm carries a 3 GHz
signal in the TE10 -mode. Calculate the maximum power handling capability of the waveguide assuming
the maximum permissible electric field inside the waveguide to be 30 kV/cm.
[GATE EC 1994 : 5 Marks]
Q.3 A rectangular hollow metal waveguide is required to be so designed to propagate a 9375 MHz signal in
its TE10 -mode that the guide-wavelength equals the cut-off wavelength. Calculate the value of a
a
(breadth or the wider dimension of the waveguide). Take b  . Also, calculate the cut-off frequency of
2
the next higher order mode. [GATE EC 1995 : 5 Marks]
Q.4 In an air - filled rectangular waveguide, the vector electric field is given by
[GATE EC 1996 : 5 Marks]
  40   
E  cos(20 y)exp   j   z  jt  iˆx V/m
  3  
Find the vector magnetic field and the phase velocity of the wave inside the waveguide.
Q.5 A rectangular waveguide with inner dimensions 6 cm  3 cm has been designed for a single mode
operation. Find the possible frequency range of operation such that the lower frequency is 5 % above the
cut off and the highest frequency is 5 % below the cut off of the next higher mode.
[GATE EC 1998 : 5 Marks]
Q.6 The region between a pair of parallel perfectly conducting planes of infinite extent in the y and z
direction is partially filled with a dielectric as shown in figure. A 30 GHz TE10 wave is incident on the
air-dielectric interface as shown. Find VSWR at the interface. [GATE EC 1998 : 5 Marks]

Q.7 A 100 m section of an air-filled rectangular waveguide operating in the TE10 mode has a cross-sectional
dimension of 1.071 cm  0.5 cm . Two pulsed carriers of 21 GHz and 28 GHz are simultaneously
launched at one end of the waveguide section. What is the time delay difference between the two pulse
at the other end of the waveguide? [GATE EC 1999 : 5 Marks]
Q.8 A rectangular hollow metal waveguide has dimensions a  2.29cm and b  1.02cm . Microwave power
at 10 GHZ is transmitted through the waveguide in the TE10 mode. [GATE EC 2001 : 5 Marks]
(a) Calculate the cut-off wavelength and the guide wavelength for this mode,
(b) What are the other ( TE or TM ) modes that can propagate through the waveguide?
(c) If a  b  2.29cm , what are the modes which can propagate through the waveguide ?
Electromagnetic Theory 184 GATE ACADEMY ®

GATE Conventional Answers

1. (i) 3.366 GHz,

(ii) 3.356 GHz
2. 23 MW
3. 2.26 cm, 13.257 GHz
1    j  z  j t  
  40   

4. H 8
cos(2y )e   3  

3 10  

5. f lowest  2.625 GHz, f highest  4.75 GHz
6. 2
7. 0.04  sec
8. (a) 3.97 cm
(b) TE10
(c) TE10 , TE01 , TE11 , TM11

3.21 IES Conventional Questions

Q. 1 Design a rectangular wave guide which at 10 GHz , will operate in TE10 mode with 25% safety factor
( f  1.25 f c ) when the interior of guide is filled with air. It is required that the mode with the next
higher cut-off will operate at 25% below its cut-off frequency. [IES EC 1993 : 17 Marks]
Q. 2 For a parallel plane waveguide, infinite in extent and a spacing of 20 cm between the plates, calculate
the [IES EC 1995 : 17 Marks]
(i) Phase velocity for TEM mode at all wavelengths.
(ii) Cutoff wavelength for dominant TE mode.
(iii) Phase velocity for dominant mode at 80% of cutoff wavelength.
(iv) The reflection angle for the above frequency of operation.
Q. 3 Determine the group velocity of a 12 GHz signal propagating in the TM 11 mode in a rectangular
waveguide of 4.0 cm  2.0 cm cross section. [IES EC 1996 : 8 Marks]
Q. 4 Find the cutoff frequency in the dominant mode of a rectangular waveguide of 2.29 cm 1.12 cm cross
section. Also find the phase velocity, guide wavelength and the impedance at 7 GHz. What is the
average power flow when the rms electric field is 800 V/m ? [IES EC 1996 : 17 Marks]
Q. 5 An air-filled rectangular waveguide of cross-section 5cm  2 cm is operating in the TE10 mode at a
frequency of 4 GHz. Determine
(i) The group velocity
(ii) The guide wavelength
(iii)The attenuation to be expected at a frequency which is 0.95 times the cut-off frequency (assuming
the guide walls to be made of perfect conductors) [IES EC 1998 : 17 Marks]
Q. 6 A wave guide has an internal breadth ‘a’ of 3 cms and carries a dominant mode of a signal of unknown
frequency of the characteristic impedance of the mode is 500 ohms. What is frequency of the signal?
[IES EC 1999 : 20 Marks]
Q. 7 Three signals (0.1 GHz, 1 GHz and 10 GHz) are available for propagation. Will it be possible to send all
of the through a parallel plate air filled waveguide with a separation of 12 cm between the plates?
[IES EC 2013 : 5 Marks]
GATE ACADEMY ® 185 Waveguides

IES Conventional Answers

1. a  1.875cm, b  1.125 cm
2. (i) 3  108 m/s
(ii) 40 cm
(iii) 5  1010 cm/s
(iv)  i  36.860
3. 2.141 cm/sec
4. 6.55 GHz, 8.5  1010 m/s, 12.22 cm, 1077 , 0.08 W
5. (i) 1.98  1010 m/sec
(ii) 11.36 cm
(iii) 0.275 Nep/cm
6. 7.62 GHz
7. Only 10 GHz will propagate.


Electromagnetic Theory 186 GATE ACADEMY ®

Space for Notes