Basic Microwave Engineering

1. MICROWAVE ENGINEERING
Introduction.
Microwave: range of frequencies included in electromagnetic spectrum.
Electromagnetic spectrum: range of frequencies for electromagnetic signals.
Electromagnetic – a form of signal energy that that combines the interaction of electric and magnetic
energy.
Electromagnetic energy results from accelerating electrons while electric energy is resultant of static
charge
𝑞₁ ⟵⟶ 𝑞₂
𝐹 =
𝑞₁𝑞₂
4𝜋𝑟²
𝐹 =
𝑞
4𝜋ɛ𝑟²
Where:
q- Charge in C.
ɛ=ɛᵣ ɛ₀.
Magnetic fields results from net magnetic moments due to spinning and revolving electrons in an atom.
Magnets have two dipoles that is N and S poles.
Magnetic diagram.
𝐻 =
𝐵
µ
, 𝐵 = µ𝐻
Where H- magnetic field density.
B –magnetic flux density.
µ- permeability of free space.

2. When the two (magnetic and electric) are combined they give rise to electro-magnetism hence
electromagnetic energy.
Electromagnetic spectrum comprises the following signals or form of energy.
 Gamma rays.
 X-rays.
 Ultraviolet light (UV).
 Infrared.
 Microwave.
 Radio wave.
Tabulate the EM spectrum.
EM Freq. wavelength Energy Application
Gamma rays 300 EHz 1 pm 1.24MeN Checking cracks in metals, building,
railway tracks..
Hard X-rays 30EHz 10pm 124KeN -medicine (cancer)
-Security –metal detectors.
Microwave
(EHF)
300GHz 1mm 1.24meV Communication –WiMAX, Radar,
-imaging.
Radio
(ELF)
3Hz 100µm 12.4µeV -communication.
-navigation.
Table.
Microwave spectrum.
Initial bands. Freq. GHz wavelength Energy Application
L 1-2 30-15 1800MHz GSM
Microwave links, waveguide.
S 2-4 15-7.5 2100MHz UMTS, WIFI, Bluetooth, Zigbee.
C 4-8 7.5-3.8 satellite
X 8-12 3.8-2.5 satellite
Ku 12-18 2.5-1.7 satellite
K 18-27 1.7-1.1 satellite
Ka 27-40 1.1-0.75
V 40-75 0.75-0.4
W 75-110 0.4-0.27
Table. Microwave spectrum.
E=hf

3. Where h-planks constant (6.62X10-38
Js).
Application of microwave.
 GSM.
 Microwave links- waveguides.
 Satellite.
 Remote sensing.
 Microwave heating.
 Radar system.
 WLANS- Wi-Fi, Bluetooth, ZigBee, WiMAX.
Differential equation.
1. Gauss’ law of electricity.
∇. 𝐸 =
𝑞
ɛ
2. Gauss law of magnetism.
∇. 𝐻 = 0
3. Faradays law
∇ × 𝐸 = −
𝜕𝐵
𝜕𝑡
4. Ampere/Maxwell law.
∇ × 𝐻=JC+JD= 𝜎𝐸 +
𝜕𝐷
𝜕𝑡
These equation expressed in free space.
∇. 𝐸 =
𝑞ₒ
ɛₒ
∇. 𝐻 = 0
∇ × 𝐸 = −
𝜕𝐵
𝜕𝑡

4. ∇ × 𝐻 =
𝜕𝐷
𝜕𝑡
Divergence.
∇=
𝛿
𝛿𝑥
+
𝛿
𝛿𝑦
+
𝛿
𝛿𝑧
Curl
∇× 𝐸 = |
âₓ â𝑦 â𝑧
𝜕
𝜕𝑥
𝜕
𝜕𝑦
𝜕
𝜕𝑧
𝐸𝑥 𝐸𝑦 𝐸𝑧
|
Determinant.
=ậx(
𝜕𝐸𝑧
𝜕𝑦
−
𝜕𝐸𝑦
𝜕𝑧
)-ậy((
𝜕𝐸𝑧
𝜕𝑥
) −
𝜕𝐸𝑥
𝜕𝑧
)-ậz(
𝜕𝐸𝑦
𝜕𝑥
−
𝜕𝐸𝑥
𝜕𝑦
)
Example.
Given.
E=Emcos(ωt + βz)
∇× 𝐸 = |
âₓ â𝑦 â𝑧
𝜕
𝜕𝑥
𝜕
𝜕𝑦
𝜕
𝜕𝑧
0 𝐸𝑦 0
|
=ậx(0 −
𝜕𝐸𝑦
𝜕𝑧
)
=-ậx
𝜕𝐸𝑦
𝜕𝑧
=-ậx
𝜕𝐸𝑚𝑐𝑜𝑠(𝜔𝑡+𝛽𝑧)
𝜕𝑧
= 𝛽Em sin(ωt + βz) ậx
But
∇ × 𝐸 = −
𝜕𝐵
𝜕𝑡

5. ∫ −
𝜕𝐵
𝜕𝑡
= ∫ βEmsin(ωt + βz)ậx
B=
𝛽
𝜔
Emcos(ωt + βz)ậx TESLA.
D=ɛO E
=ɛO Emcos(ωt + βz)
𝐶
𝑀²
H=
𝐵
µ
=
𝛽
𝜔µ
Emcos(ωt + βz) ậx
𝐴
𝑀
.
Impedance.
Z=
𝐸
𝐻
.𝜴
=
𝜔µ
𝛽
𝜴.
Appling the same theorem, find B, E and D. given.
H=HM 𝑒(𝜔𝑡−𝛽𝑧)
ậx .
Exercise.
Given
H=20𝑒 𝑗(109 𝑡+𝛽𝑧)
ậy
𝑚𝐴
𝑀
.
Determine D, H, and B then draw the waveform.
For the analysis above obtain the actual values in the microwave range. (Microwave range 300MHz-
300GHz), but 1-10GHz are commonly used.

6. Electromagnetic wave analysis based on Maxwell’s equation in time varying form in space.
∇ × 𝐸 = −𝑗𝜔µ𝐻
∇ × 𝐻 = 𝜎𝐸 + 𝑗𝜔ɛ𝐸
∇. 𝐵 = 0
∇. 𝐷 = 0
Where:
B=µH, J= 𝜎E, D=ɛE.
Taking the curl of both expressions.
The curl of E.
∇ × (∇ × 𝐸) = −𝑗𝜔µ(𝛻 × 𝐻)
= −𝑗𝜔µ(𝜎𝐸 + 𝑗𝜔ɛ𝐸)
From curl identity, this can expressed as,
𝛻2
𝐸 = 𝑗𝜔µ(𝜎𝐸 + 𝑗𝜔ɛ𝐸)
= 𝑗𝜔µ(𝜎 + 𝑗𝜔ɛ)𝐸
The curl of H.
∇ × (∇ × 𝐸) = 𝛻 × (𝜎𝐸 + 𝑗𝜔ɛ𝐸)
= 𝜎𝛻 × 𝐸 + 𝑗𝜔ɛ𝛻 × 𝐸
𝛻2
𝐻 = 𝜎(−𝑗𝜔µ𝐻) + 𝑗𝜔ɛ(−𝑗𝜔µ𝐻)
= 𝑗𝜔µ(𝜎 + 𝑗𝜔ɛ)𝐻
By taking, =𝛻, where 𝛶-gamma. We have.
Υ2
𝐻 = 𝑗𝜔µ(𝜎 + 𝑗𝜔ɛ)𝐻
Υ2
= 𝑗𝜔µ(𝜎 + 𝑗𝜔ɛ)
𝛶 = √(𝑗𝜔µ(𝜎 + 𝑗𝜔ɛ))
Or
𝛶2
𝐸 = 𝑗𝜔µ(𝜎 + 𝑗𝜔ɛ)𝐸
𝛶2
= 𝑗𝜔µ(𝜎 + 𝑗𝜔ɛ)
𝛶 = √(𝑗𝜔µ(𝜎 + 𝑗𝜔ɛ))
𝛶 = 𝛼 + 𝑗𝛽

7. 𝛼 = 𝜔√(
µɛ
2
((√1 + (
𝜎
𝜔ɛ
)
2
) − 1)
𝛽 = 𝜔√(
µɛ
2
√𝐻 (
𝜎
𝜔ɛ
) ² + 1)
The above equation are for partially medium.
Perfect dielectric
For a perfect dielectric.
𝜎=0,
𝛼=0,
𝛽=𝜔√µ𝜔
Perfect conductor.
For a perfect conductor.
𝜎 >>𝜔ɛ. Which can be expressed as (
𝜎
𝜔ɛ
) ² >> 1.
𝛼 = 𝛽 = 𝜔√(
µɛ
2
.
𝜎
𝜔ɛ
)
= 𝜔√
µ𝜎
2𝜔
= √
𝜔µ𝜎
2
But 𝜔=2𝜋f.
Therefore.
= √𝜋𝑓µ𝜎
Skin depth of a conductor, which is how far can signal penetrate into a conductor, is expressed in the
form.
𝛿 =
1
𝛽
=
1
√𝜋𝑓µ𝜎

8. Exercise.
Determine the microwave skin depth of a waveguide F=10GHz, µr=2 (Cu, Al, Fe). Also find the mostly
used material for waveguides and why.
Effects on EM as they propagate through media.
 Reflection
 Refraction
 Attenuation
 Scattering
 Diffraction
 Absorption
 Depolarization
 Fading
 Losses
 Dispersion
Reflection.
Bouncing back when they are in contact with an obstacle. In free space this can be buildings, vehicles,
flying objects, mountains etc. in waveguide transmission is done by reflection, metallic objects are
mostly used because are considered perfect reflectors.
Obeys laws of reflection, depends on µ and ɛ but mostly permittivity.
Related by:
sin 𝜃₁
sin 𝜃₂
= √(
ɛ₂µ₂
ɛ₁µ₁
)
For free space, (air) µ₁=µ₂=1
Refraction.
This bending of waves during propagation as they travel from one medium to the other. Obeys
Snell’s law.
𝑛₁ sin 𝜃₁ = 𝑛₂ sin 𝜃₂
Where n1 and n2 are the refractive indexes.
Refractive index, n, depends on the permittivity of the medium.
In waveguides different air density can affect the signal. Fiber as a waveguide has material with different
n.

9. Attenuation.
Process when the wave losses power/energy as it propagates. Depends on attenuation constants, which
also depends on material type/medium.
𝛶 = 𝛼 + 𝑗𝛽
↕
E=EM 𝑒 𝛶𝑡+𝑗𝜔𝑧
=EM 𝑒 𝑗(𝛶𝑡+𝛽𝑧)
where 𝛶=j𝛽
=EMcos(𝜔𝑡 + 𝛽𝑧) 𝑜𝑟 E=EMsin(𝜔𝑡 + 𝛽𝑧)
Affected by frequency.
Scattering
Reflection from non-uniform surfaces. In contact with particles with wavelength close to the
surface/object size. In wave guides it can be caused by minute particles and irregular surface.
Diffraction.
When the waves comes into contact with a sharp object/edge. It’s observed at the ends of the
waveguides.
Diagram.
Depends on the waveguides of the EM wave.
Dispersion.
Signal/wave spreads as it propagates making the signal weaker or cancellation sometime signal addition.
Depolarization
Polarization orientation of the EM wave. There are three types
 Linear (vertical and horizontal )
 Circular
 Elliptical
Depolarization- change of alignment vertical to horizontal or viceversa.in waveguide, its cause variation
of ionization level due to density variation.
Absorption.

10. Transfer of energy to obstacles or medium material resulting in change to other form of energy e.g.
heat.
Exercise: discuss the difference between absorption and attenuation.
Fading
Reduction of signal strength due to variations as it propagates.
Losses.
 Free space losses.
 Feeder losses due to misalignment.
 Obstacle losses.
Passive microwave devices.
Passive devices do not require external power source to operate e.g. resistor. Do not increase the
strength of the signal. Most waveguides are in this category.
A waveguide is a device that directs an EM wave through it as it move from source to destination.
EM waves directed includes:
 TEM- Transverse Electric and Magnetic wave
 TE -Transverse Electric wave
 TM- Transverse magnetic wave.
TEM. Has both magnetic and electric properties as components of the wave being propagated.
TE- has electric energy only.
TM. Has the magnetic energy only.
The wave results in different modes of propagation through the waveguides. They can be analyzed and
determined using Maxwell’s equations. The equation are given by: Faradays’ law of magnetism and
Amperes’ law of electricity.
Using Faraday’s law.
Note all equation are in vector form.
∇ × 𝐸 = −𝑗𝜔µ𝐻
∇× 𝐸 = |
âₓ â𝑦 â𝑧
𝜕
𝜕𝑥
𝜕
𝜕𝑦
𝜕
𝜕𝑧
𝐸𝑥 𝐸𝑦 𝐸𝑧
|=−𝑗𝜔µ𝐻
ậx:(
𝜕𝐸𝑧
𝜕𝑦
−
𝜕𝐸𝑦
𝜕𝑧
) = −𝑗𝜔µ𝐻x
ậy:(
𝜕𝐸𝑥
𝜕𝑥
−
𝜕𝐸𝑧
𝜕𝑧
)=𝑗𝜔µ𝐻y

11. ậz:(
𝜕𝐸𝑦
𝜕𝑥
−
𝜕𝐸𝑥
𝜕𝑦
) =−𝑗𝜔µ𝐻z
Taking the waveform in z-direction to be given by 𝑒−𝑗𝐵𝑍
and substituting in the above equation.
ậx.(
𝜕𝐸𝑧
𝜕𝑦
−
𝜕𝐸𝑦
𝜕𝑧
) = −𝑗𝜔µ𝐻x
−𝑗𝜔µ𝐻x =
𝜕𝐸𝑧
𝜕𝑦
+ 𝑗𝐵𝐸y
𝑗𝜔µ𝐻y =
𝜕𝐸𝑧
𝜕𝑧
+ 𝑗𝐵𝐸x
ậz:(
𝜕𝐸𝑦
𝜕𝑥
−
𝜕𝐸𝑥
𝜕𝑦
) =−𝑗𝜔µ𝐻z
Using Amperes equation.
∇ × Ĥ = 𝑗𝜔µ𝐻
∇× 𝐻 = |
âₓ â𝑦 â𝑧
𝜕
𝜕𝑥
𝜕
𝜕𝑦
𝜕
𝜕𝑧
𝐻𝑥 𝐻𝑦 𝐻𝑧
|=𝑗𝜔ɛ𝐸
ậx :(
𝜕𝐻𝑧
𝜕𝑦
−
𝜕𝐻𝑦
𝜕𝑧
) = 𝑗𝜔µ𝐸x
𝜕𝐻𝑧
𝜕𝑦
−
𝜕𝑒−𝑗𝐵𝑍
𝜕𝑧
= 𝑗𝜔µ𝐸x
𝜕𝐻𝑧
𝜕𝑦
+ 𝑗𝐵𝐻y= 𝑗𝜔µ𝐸x
ậy :(
𝜕𝐻𝑧
𝜕𝑥
−
𝜕𝐻𝑥
𝜕𝑧
)=𝑗𝜔µ𝐸y
𝜕𝑒−𝑗𝐵𝑍
𝜕𝑥
−
𝜕𝐻𝑥
𝜕𝑧
= 𝑗𝜔µ𝐸y
=
𝜕𝐻𝑧
𝜕𝑧
+ 𝑗𝐵Hx
ậz :(
𝜕𝐻𝑦
𝜕𝑥
−
𝜕𝐻𝑥
𝜕𝑦
) =𝑗𝜔µ𝐸z
Manipulating the above equation. We can get the transverse components of Ē and Ĥ, i.e. the x and y
components.
Taking the first expression and making Hx the subject of the formula we obtain.

12. Hx =− (
𝜕𝐸𝑧
𝜕𝑦
+𝑗𝐵𝐸y
𝑗𝜔µ
)
=
𝑗
𝜔µ
(
𝜕𝐸𝑧
𝜕𝑦
+ 𝑗𝐵𝐸y)
=
𝑗
𝜔µ
𝜕𝐸𝑧
𝜕𝑦
−
𝐵𝐸y
𝜔µ
But.
Ey=
𝜕𝐻𝑧
𝜕𝑥
+
𝛽
𝜔ɛ
Hx
Therefore.
Hx=
𝑗
𝜔µ
𝜕𝐸𝑧
𝜕𝑦
−
𝐵
𝜔2µ
𝜕𝐻𝑧
𝜕𝑥
+
𝛽²
𝜔²µɛ
Hx
=
(
𝑗
𝜔µ
𝜕𝐸𝑧
𝜕𝑦
−
𝐵
𝜔2µ
𝜕𝐻𝑧
𝜕𝑥
)
1−
𝛽²
𝜔²µɛ
Let, 𝜔²µɛ= k²
Hx =
𝑗
k²−β²
(𝜔ɛ
𝜕𝐸𝑧
𝜕𝑦
− 𝛽
𝜕𝐻𝑧
𝜕𝑥
)
And kC²= k²-𝛽²
Using the same procedure obtain: Hy, Ex and Ey
Hy =−
𝑗
kc²
( 𝜔ɛ 𝜕𝐸𝑧
𝜕𝑥
+ 𝛽 𝜕𝐻𝑧
𝜕𝑦
)
Ex =−
𝑗
kc²
( 𝛽 𝜕𝐸𝑧
𝜕𝑥
+ 𝜔µ 𝜕𝐻𝑧
𝜕𝑦
)
Ey =
𝑗
kc²
(−𝛽 𝜕𝐸𝑧
𝜕𝑦
+ 𝜔µ 𝜕𝐻𝑧
𝜕𝑥
)
All the above equation are TEM such that the electric and magnetic exist in z-direction, thus longitudinal
with the direction of propagation and therefore both of them exist i.e. EZ and HZ.
Modes
TE Mode.

13. Transverse electric: indicates that electric field is cross with direction of propagation hence EZ=0 but
HZ≠0.
The above expressions becomes:
Hx =
𝑗
kc²
(−𝛽
𝜕𝐻𝑧
𝜕𝑥
)
Hx =−
𝑗
kc²
𝛽 𝜕𝐻𝑧
𝜕𝑥
Ex =−
𝑗
kc²
( 𝜔µ 𝜕𝐻𝑧
𝜕𝑦
)
=−
𝑗
kc²
𝜔µ 𝜕𝐻𝑧
𝜕𝑦
Ey =
𝑗
kc²
( 𝜔µ 𝜕𝐻𝑧
𝜕𝑥
)
=
𝑗
kc2 𝜔µ 𝜕𝐻𝑧
𝜕𝑥
Hy =−
𝑗
kc²
𝛽 𝜕𝐻𝑧
𝜕𝑦
TM Mode.
Transverse magnetic: the magnetic fields is cross to the direction of propagation, hence EZ≠0 but
HZ=0.
The above expression becomes.
Hx =
𝑗
kc²
𝜔ɛ
𝜕𝐸𝑧
𝜕𝑦
Hy =−
𝑗
kc²
𝜔ɛ 𝜕𝐸𝑧
𝜕𝑥
Ex =−
𝑗
kc²
𝛽 𝜕𝐸𝑧
𝜕𝑥
Ey =−
𝑗
kc²
𝛽 𝜕𝐸𝑧
𝜕𝑦
TEM Mode
The E and H are both transverse i.e. perpendicular to the direction of propagation, both EZ and HZ
are zero. HX, HY, EX and Ey do not exist.

14. The waveguides that can propagate TEM should have two lines for instance coaxial, microstrip and
coplanar.
Types of waveguides.
There are several types classified according to the following features.
 Number of lines.
 Material –conductor, dielectric.
 Types of modes of propagation TE, TM, TEM, and hybrid.
 Cut of frequency.
 Shape: Rectangular, Cylindrical and Elliptical.
 Material type. Level of permittivity.
 Technology.
Based on the above, the types includes:
 Metallic waveguides.
 Metallic TEM and quasi TEM.
 Dielectric.
Metallic waveguides.
Are constructed from metallic material, are in different shapes including:
1. Rectangular.
Supports TE and TM but not TEM. It’s has a cut off frequency.
2. Cylindrical.
The face is circular with a given length, radius is the main consideration in its design.
R
Diagram.
Supports TE and TM.
3. Elliptical.
Width

15. Diagram.
Supports TE and TM.
Quasi TEM.
Have more than a single line. Includes:
 Coaxial,
 Strip-line
 Coaxial rectangular.
Coaxial.
Inner core.
Outer conductor
insulator
Diagram
It has an inner core, conductor that is insulated and separated from the outer conductor. At the top
there is an insulator referred as outer jacket. The two radii, inner and outer radii, are considered is its
design.
Coaxial rectangular.
Has rectangular dimensions.

16. Inner core.
Outer conductor
dielectric
Diagram.
Strip-line
Construction.
Ground
w
b
ɛᵣ
Strip-line
Diagram.
Constructed by etching a conductor with width of W on a dielectric material with ɛr in the middle of the
dielectric as shown above.
The most important parameter are the width of the conductor (w) and the height of the dielectric (b).
The conductor appears at
𝑏
2
distance from either side upper or lower. This parameter b and w will
directly affect the capacitive and inductive properties of the line which together with ɛr also affects the
plane, group velocities, propagation constant and the impedance of the device.
Vg =
𝑐
√ɛᵣ
c- Velocity of light.
The fields will appear as follows.

17. Diagram.
It supports TEM mode.
Other lines include.
 Slot lone
 Coplanar.
 Microstrip.
Microstrip line.
microstrip
Diagram.
Constructed by etching a conductor at the top of a dielectric material with a ground running across the
bottom of the dielectric.
The important parameter include w-width of the line, b the separation between the ground and the line
and the dielectric material constant ɛr. These will affect capacitive and inductive properties as well as vp,
vg, 𝜂 and 𝛽.
The fields are given by
Diagram.
Impedance matching two port network.
NW
2
NW
1
High
impedance
Low
impedance
Connecting in this form causes reflection of signal, it should be connected as below.

18. NW
2
NW
1
High
impedance
Low
impedance
Transistor can be used for impedance matching.
Covered microstrip.
Similar to the microstrip the difference being that strip is covered.
Cover (dielectric
material) microstrip
Diagram.
The cover has an effect which can be negative in that it can introduce losses.
Slot line.
Construction.
Slot line
Diagram.
Constructed by etching a metallic material at the top of the dielectric creating a section without the
conductor, this forms the slot.
Coplanar.

19. Construction.
Has two slots.
Diagram.
Obtained by etching two sessions on a metal conductor then coating done on dielectric.
Additional devices.
 Ridge waveguides.
 Dielectric waveguides.
i) Fiber waveguide.
ii) Plain dielectric.
Ridge waveguide.
Ridges are developed by a dielectric material.
Ridges (ɛᵣ₁ )
Diagram
Ridges are developed on a dielectric material.
Fiber waveguide.
Construction.
ɛ₂
ɛ₁

20. Diagram.
It developed from two materials with different dielectric properties.
Reflection and refraction can be principles can be applied in their applications.
Plain dielectric.
ɛᵣ
Diagram.
Developed using single material type with specific dielectric properties. Supports TE and TM.
Active Microwave devices.
Require power to operate and can increase the strength of a signal actually includes electronics devices
semiconductor, Diodes, Transistors, Integrated circuits.
Diodes.
Types:-
 Schotty diode.
 PIN diode.
 Tunnel diode.
Transistors include BJT, FETs.
Integrated circuits includes:
 BJT based technology
 FET based technology.
Others are; semiconductor resistor and capacitor.
Exercise.
1. Derive the cut-off frequency for all the above waveguides discussed. Obtain: vp, vg, impedance (𝜂),
propagation constant (𝛽), for all the waveguide above.

21. 2. Discuss the different active microwave devices. Discussion should include: definition, symbol,
construction, features, operation and applications.
3. Do the same for passive microwave devices.