ELEC 401 – Microwave Electronics
ELEC 401
MICROWAVE ELECTRONICS
Lecture 1
Instructor: M. İrşadi Aksun
Acknowledgements:
1. Two animations on Faraday’s law were taken from the following web page:
http://web.mit.edu/jbelcher/www/inout.html
2. Artworks used to discuss Faraday’s law, Ampere’s law, Gauss’s law were
taken from the following web page:
http://cobweb.ecn.purdue.edu/~ece695s/Lectures
M. I. Aksun
Koç University
1/17
ELEC 401 – Microwave Electronics
Outline
 Chapter 1: Motivation & Introduction
 Chapter 2: Review of EM Wave Theory
 Chapter 3: Plane Electromagnetic Waves
 Chapter 4: Transmission Lines (TL)
 Chapter 5: Microwave Network Characterization
 Chapter 6: Smith Chart & Impedance Matching
 Chapter 7: Passive Microwave Components
M. I. Aksun
Koç University
2/17
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
 Governing equations of EM waves are:
- Maxwell’s Equations in integral form:
 ~
~
 E  dl    B  ds
t A
C
 ~
~
~
H

d
l

D

d
s



 J  ds
t A
C
A
~
~
D
  ds    dv
A
V
~
B
  ds  0
~
E(r, t )
~
H(r , t )
~
D(r, t )
~
B(r, t )
~
J(r, t )
~
(r, t )
Electric Field Vector [V/m]
Magnetic Field Vector [A/m]
Electric Flux Density [C/m2]
Magnetic Flux Density [W/m2]
Current Density [A/m2]
Charge Density [C/m3]
A
 ~
~
- Continuity equation  J  ds     dv
t V
A
M. I. Aksun
Koç University
3/17
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
- Maxwell’s Equations in differential form: How do you get
them from their integral forms?
~
B
~
E  
(Faraday-Maxwell law)
t
~
~ D ~
H 
 J (Generaliz ed Ampere' s law)
t
~
D  ~
 (Gauss's law)
~
  B  0 (Law of conservati on of magnetic flux)
~
~
- Continuity equation
J   
t
Boldface and  are used throughout this course to represent vector and timevarying forms of the corresponding quantities, respectively
M. I. Aksun
Koç University
4/17
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
 Faraday-Maxwell Law
 ~
~
 E  dl    B  ds
t A
C
M. I. Aksun
Koç University
5/17
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
 Faraday-Maxwell Law
M. I. Aksun
Koç University
6/17
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
 Faraday-Maxwell Law
M. I. Aksun
Koç University
7/17
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
 Generalized Ampere’s Law
 ~
~
~
 H  dl   D  ds   J  ds
t A
C
A
M. I. Aksun
Koç University
8/17
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
 Gauss’ Law
 D  ds   dv
A
V
 B  ds  0
A
M. I. Aksun
Koç University
9/17
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
 What are the Maxwell’s Contributions?
1. Interpretation of Faraday’s Law
 ~

~
E.M.F   E  dl    B  ds  
t A
dt
C
C: Conducting Loop
No need for a conducting loop to induce electromotive force
M. I. Aksun
Koç University
10/17
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
 What are Maxwell’s Contributions?
2. Correction of Ampere’s Law: Physical
Original Ampere’s law
~
~
H

d
l


 J  ds
C
A
 ~
~
~
H

d
l

D

d
s



 J  ds
t A
C
A
Maxwell’s contribution
M. I. Aksun
Koç University
11/17
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
 What are the Maxwell’s Contributions?
2. Correction of Ampere’s Law: Mathematical
~  ~ ~)
  (  H  D  J
t
~ ~)
(H
J
~ 
~
~


H 
DJ





t
~
~


H  J




0

~
~
J   
t
0
~
J  0
Conservation of Charges
M. I. Aksun
Koç University
12/17
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
 Phenomenological Picture of Wave Generation
E (t )
H (t )
E (t )
H (t )
I (t )
(a)
H 

DJ
t
M. I. Aksun
Koç University
E (t )
E (t )
H (t )
H (t )
I (t )
(b)
I (t )
E  

B
t
H (t )

H  D
t
(c)
13/17
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
 Time-harmonic representations of Maxwell’s Equations:
Maxwell’s equations are linear equations, as they involve linear operators
like integrals or derivatives,
Medium is assumed to be linear with linear relations between B and H, and
D and E,
Therefore,
Time varying sources can be written in terms of pure sinusoids (harmonics)
via Fourier transform,
Superposition principle can be applied since the equations and the involved
media are all linear,
M. I. Aksun
Koç University
14/17
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
As a result, all involved field quantities have the same frequency of
oscillations, and can be written as follows:












~
E x, y, z , t   Re E x, y, z e j t
~
H  x, y, z , t   Re H  x, y, z e j t
~
D x, y, z , t   Re D x, y, z e j t
~
B x, y, z , t   Re B x, y, z e j t
~
J  x, y, z , t   Re J  x, y, z e j t
~  x, y, z , t   Re   x, y, z e j t
For example: An x-polarized electric
field may be given in time-domain as
~
Ex, y, z, t   xˆ Ex, y, z cost   
where all quantities are real.
It can also be written in time-harmonic
form (phasor form, frequency domain)
as a complex quantity:
E x, y, z   xˆ E  x, y, z e j
Complex quantities
M. I. Aksun
Koç University
15/17
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
Let us apply this representation to Generalized Ampere’s law:
~
D(r, t ) ~
~
  H (r, t ) 
 J (r, t )
t

j t
  e{H (r )e }  e{D(r )e jt }  e{J (r )e jt }
t
  H (r )  jD(r )  J (r )
and the rest become
  E(r )   jB(r )
  D(r )   (r )
  B(r )  0
M. I. Aksun
Koç University
16/17
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
As a result, we have the following sets of Maxwell’s equations
in
Differential form
Integral form
 E  dl   j B  ds
C
  E   jB
A
 H  dl  j D  ds   J  ds
C
A
A
 D  ds   dv
A
  H  jD  J
D  
V
B  0
 B  ds  0
A
 J  ds   j  dv
A
M. I. Aksun
Koç University
  J   j
V
17/17