# Microwaves Elec401 Lec3

```ELEC 401 – Microwave Electronics
ELEC 401
MICROWAVE ELECTRONICS
Lecture 3
Acknowledgements:
1. Animation on the visualization of EM waves was taken from the following
web page:
http://web.mit.edu/~sdavies/MacData/afs.course.lockers/8/8.901/2007/Tues
dayFeb20/graphics/
M. I. Aksun
Koç University
1/18
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
2/18
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
 Wave equations are the governing differential equations of
EM waves
 ~
 E   2 E  0
t
2~
 2E  2E  0
 ~
2~
 H   2 H  0
t
2
2
Wave equation
Helmholtz equation
2H  2H  0
Facts: Wave equations
- have solutions representing wave nature of the fields;
- are second order partial differential equations;
- have unique solutions if two boundary conditions are introduced.
So, we need BOUNDRY CONDITIONS
M. I. Aksun
Koç University
3/18
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
 Boundary Conditions on E and H:
 E  dl   j  B  ds
C
D1 ,B1
̂
n̂
n̂1
Pill-box
n̂
̂
ds
S
D 2 ,B 2
 2 , 2
 H  dl  j  D  ds   J  ds
C
l
S
S
H1  τˆl  H2  τˆl  jωD  nˆ1l h  J  nˆ1l h
lim J  nˆ1h  nˆ1J s
Contour C
E2 ,H 2
Δh 0
as h0
M. I. Aksun
Koç University
E1  E2
as h0
Medium 2:
h
E1  τˆl  E2  τˆl   j B  nˆ1l h

Medium 1: 1 , 1
E1 , H1
S
A/m
H1  H 2  J s
4/18
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
Comments on the surface current density Js :
lim J  nˆ1h  nˆ1J s
Δh 0
Medium 1:  1 , 1
Volume current density must go to
as h goes zero.
Conducting
Medium 2:
Js

J s (0)e
z
M. I. Aksun
Koç University
A/m
z

 2 ,  2 ,

Current Sheet: Current is
distributed over a very narrow
sheet of conductor.
Facts: 1. Current flow in a good conductor is
practically confined to the layer next to the surface,
whose thickness depends inversely on both the
conductivity of the material and the frequency of the
field.
2. Conductor is modeled as a sheet of current with a
finite surface current density.
3. This layer of finite thickness is called the skin
depth of that specific conducting material.
5/18
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
 Boundary Conditions on D and B:
 D  ds    dv
S
D1 ,B1
̂
n̂
n̂1
Pill-box
n̂
̂
Medium 1: 1 , 1
Medium 2:  2 , 2
h
D 2 ,B 2
l
D1  nˆ S  D2  nˆ S  sw  ρ S Δh
lim  h   s
S
E1 , H1
V
h 0
C/m 2
as h0 D1n  D2n   s
Similarly, from
 B  ds  0
S
Contour C
E2 ,H 2
M. I. Aksun
Koç University
B1n  B2 n
6/18
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
 Let us review the boundary conditions with words:
1. Tangential E fields are continuous
D1 ,B1
̂
n̂
Pill-box
n̂1
n̂
̂
Medium 1: 1 , 1
S
Medium 2:
E1 , H1
h
D 2 ,B 2
l
 2 , 2
E1  E2
2. Tangential H fields are discontinuous
by the amount of the surface current
H1  H 2  J s
3. Normal components of D are discontinuous
by the amount of the surface charge
D1n  D2 n   s
Contour C
E2 ,H 2
4. Normal components of B are continuous
B1n  B2 n
M. I. Aksun
Koç University
7/18
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
 Special case: Dielectric-PEC interface (   )
1. Tangential E field is zero
E1  0
Medium 1:  1 , 1
Conducting
Medium 2:
Js

J s (0)e
z
z

 2 ,  2 ,
2. Tangential H field is equal to the
surface current density
H1  J s
3. Normal component of D is equal to the
surface charge density
D1n  s
4. Normal component of B is zero
B1n  0
M. I. Aksun
Koç University
8/18
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
 So far, we have derived the governing equation of the fields
involved in waves, and the necessary boundary conditions;
 Now, it is time to see them in action to determine the fields
in a wave generated by a time-varying current source !!!!
M. I. Aksun
Koç University
9/18
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
 Example: Assume an infinite sheet of electric surface current density J s  xˆJ 0
A/m is placed on a z  0 plane between free space for z < 0 and a dielectric medium
with for z > 0. Find the resulting electric and magnetic fields in both regions.
x
0
   0 r
x̂J 0 A/m
1. The surface current density is
uniform on the z = 0 plane;
z
2. Magnitude of the current density is
j t
J0 with e
time dependence;
3. We assume that layers are lossless,
isotropic, homogenous, and semiinfinite in extent.
M. I. Aksun
Koç University
10/18
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
The best way to solve the wave equation in a piecewise homogeneous
geometry is
1. to find the source-free solutions with unknown coefficients in each
homogeneous sub-region;
2. to apply the necessary boundary conditions at the interfaces to account
for the boundaries between different media; and then
3. to apply the boundary conditions at the sources to incorporate the
influence of the sources into the solution.
M. I. Aksun
Koç University
11/18
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
So, let us implement these steps one-by-one:
1. Write the frequency-domain wave equation in a source-free medium as
2 E  ω2 με E  0
From the source distribution and geometry, the functional form of the
solutions can be predicted.
E  xˆE x z   yˆ E y z   zˆE z z 
H  xˆH x z   yˆ H y z   zˆH z z 
Source and geometry are independent of x and y, and so are the solutions!!!
WHY ?
M. I. Aksun
Koç University
12/18
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
As we have noted earlier, the field components must satisfy Maxwell’s
equations, so using



 0,
 0,
0
x
y
z
Maxwell’s equations can be written as follows:
  E   xˆ


E y  yˆ E x   jH x xˆ  jH y yˆ
z
z
  H   xˆ


H y  yˆ H x  jωE x xˆ  jωE y yˆ
z
z
As a result, two decoupled set of solutions are obtained:
Ex
  jH y ; 0  H z
z
2Ey
H y
H x
 jE y ;
  jEx ; 0  Ez
z
z
 2 Ex
E y
z
 jH x ;
M. I. Aksun
Koç University
z
z
2
2
 2E y  0
E y , H x 
 2E x  0
Ex , H y 
13/18
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
- Note that these solutions are possible solutions; either only one of them or
both with different weights can exist depending upon the boundary
conditions.
x
- Since the surface current density is in xdirection, it gives rise to a discontinuity
only on Hy .
0
   0 r
z
x̂J 0 A/m
- Therefore, it can be interpreted that the
source excites only one of the two possible
solutions, Ex , H y  with the following
governing equation:
 2 Ex
z 2
 2E x  0
M. I. Aksun
Koç University
  0 z < 0
k0  200 z < 0
   0 r z > 0
k  20r 0 z > 0
14/18
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
- Then, the solutions can be written as
E x z   Ae jk0 z  Be jk0 z , for z < 0
E x z   Ce jkz  De jkz , for z > 0
where A, B, C, D are unknown coefficients, and k0 and k are the wavenumbers in
free-space and materials, respectively.
- The corresponding magnetic field Hy can be obtained from FaradayMaxwell equations as follows:

E x z    jH y z 
z
H y z  
H y z  
M. I. Aksun
Koç University
k0
Ae jk0 z 
k
Ce jkz 


k0
Be jk0 z , for z < 0
k
De jkz , for z > 0


15/18
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
- There are four unknowns (A, B, C and D) to be determined by using the
necessary boundary conditions:
1. Source is at z=0, and no wave can travel from infinity towards
source, implying that A and D must be zero.
 Be , for z < 0
Ex z     jkz
Ce , for z > 0
jk0 z
 k0
jk0 z
  Be , for z < 0
H y z   
k

Ce jkz , for z > 0
 
2. The remaining boundary conditions are at the interface:
E x z  0    E x z  0  
H y z  0   H y z  0   J 0
M. I. Aksun
Koç University
BC
B
 J 0
k0  k 
16/18
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
 Energy Flow and Poynting’s Vector
-The general law of the conservation of energy states that
if an object radiates electromagnetic waves (for example,
light), it loses energy.
- From Maxwell’s equations, one can get the following
relation:
~ ~
~ ~  1 ~ ~ 1 ~ ~
  E  H  ds   E  J     E  E   H  H dV
t V  2
2
V
S
 

 

Inflow of energy
due to source
outsideV
Energy flux
density (W/m2)
M. I. Aksun
Koç University
Energy
dissipated
in volume V
Rate of increase in stored energy
~
~
~




P r, t  E r, t  Hr, t 
Instantaneous
Poynting’s vector
17/18
ELEC 401 – Microwave Electronics
Review of EM Wave Theory
 For most practical applications, the time-averaged energy and power
quantities are required because most systems usually respond to the
average power, rather than its instantaneous values.
T
1 ~
Pr    Pr, t  dt : time-averaged power density
T 0
For time-harmonic fields, the instantaneous fields are written as




1
~
Er, t   ReEr e j t  Er e j t  E r e  j t
2
1
~
Hr, t   ReHr e j t  Hr e j t  H r e  j t
2

2
2
1 1 
j 2 t


 j 2 t


Pr  
dt  
E
H

E

H
E  H   e dt  E  H   e

2π 4 
0
0
2 Re E  H  







0
0

M. I. Aksun
Koç University



0 dt  


2

1
Re E  H*
2

18/18
```