Download Determination of Relative Dielectric constant and Thickness (1)

Millimeter Wave Sensor: An Overview
SURYA K PATHAK
INSTITUTE FOR PLASMA RESEARCH
BHAT, Gandhinagar – 382428 , GUJARAT, INDIA
[email protected]
Interaction of Electromagnetic Waves with Material Medium
Interaction produces
following Physical
Phenomenon which
are basis of Material
characterization
•Reflection
•Refraction
•Attenuation
•Scattering
•Diffraction
Techniques for Material
measurements
Time delay measurements
Phase measurements
Inverse Scattering methods
Interferometry and Reflectometry systems
Interferometry and Reflectometry is a
scientific technique, to interfere or correlate
two, or more than two, signals to from a
physically observable measure, from which
any useful information can be inferred.
Measurement
Mirror
Displacement
Beam Splitter
- Fringe pattern in optical Interferometry
- Electrical signals of power/ voltage in
Radio measurement
Light Source
Michelson Interferometer
Reference Mirror
Screen/ Detector
Measurement
Path
Measurement
Path
Antenna
Frequency
Source
Circulator
Antenna
Frequency
Source
Power
Divider
Plasma or Dielectric
Plasma or
Dielectric
Power
Divider
Quadrature Mixer
Reference
Path
Interferometer system
Reference
Path
I
Q
Reflectometer system
Quadrature
Mixer
I
Q
Interaction of Electromagnetic wave dielectrics (1)
The electromagnetic wave incident normally on the
dielectric and traveling in z-direction is given by:
Ei ( z, t )  E0e( jt  z )
(1)
Where  is the propagation constant. On applying
boundary conditions across the individual
boundaries. Following expressions are obtained in
terms of reflected Ei and transmitted Er wave:
E0 (1   o )  E1 (1   1 )
E0
E
(1   0 )  1 (1   1 )
Z0
Z1



0
1
2
0
1
2

y


Ei
Et
z
(2)
Er
E1 (e1d   1e 1d )  T0 E0
E
E1 1d
(e   1e 1d )  T0 0
Z1
Z2
Where 0, and T0 are reflection and transmission
coefficient.
Z is the impedance of the medium
Subscript corresponds to respective medium
Dielectric
0
d
Interaction of Electromagnetic wave dielectrics (2)
Solving above equations we can calculate the components of reflected and transmitted wave:
The reflected wave is obtained by
Er   0 E0
where,
(3)
 Z1 
 Z1 Z 2 

1
cosh(

d
)



   sinh( 1d )
1
Z0 
Z 0 Z1 


0 
.
 Z1 
 Z1 Z 2 

1
cosh(

d
)



   sinh( 1d )
1
 Z0 
 Z 0 Z1 
And the transmitted wave is expressed by
Et  T0 E0
where,


 Z1 Z 0 
 1  Z 0 

T0   1   cosh( 1d )     sinh( 1d )  
 Z 2 Z1 
 2  Z 2 



(4)
1
Wave equations (3) and (4) constructs the measurement path wave in an interferometry
Determination of Relative Dielectric constant and Thickness (1)
Complex Permittivity or Dielectric Constant
The relative permittivity or dielectric constant (r) as well as relative permeability (r)
characterize the relationship between electromagnetic waves and material properties. For a
lossy material, the relative dielectric constant can be expressed as:



 r   r  j
0
 0

(5)
where r is the relative dielectric constant, 0 is the dielectric constant of free space,  is the
conductivity of the material, and  is angular frequency of EM waves. Above equation can be
represented in terms of loss tangent (which is ratio of imaginary to real component of the
complex dielectric constant) :

(6)
 r   r (1  j tan  )
Reflection Measurement
In reflection measurement method, the dielectric is typically conductor-backed to increase the
reflected power. In this case, Z2 is equal to zero and reflection coefficient in equation (3) is:
( 0   1 )e1d  ( 0   1 )e1d
0 
( 0   1 )e1d  ( 0   1 )e1d
(7)
Determination of Relative Dielectric constant and Thickness (2)
Impedance Z1 in terms of free space impedance is given by:
Z1 
Z0

 r1

0
Z
1 0
(8)
Transmission Measurement
In this method, the dielectric is located in free space between two antennas. Therefore, Z2 =
Z0 and 2 = 0 are satisfied, and transmission coefficient is given as:
4 1 0
T0 
( 0   1 ) 2 e 1d  ( 0   1 ) 2 e1d
(9)
Determination of Relative Dielectric constant and Thickness (3)
Fig: Phase of reflection coefficient depending on (a) relative dielectric constant
(b) dielectric thickness
Determination of Relative Dielectric constant and Thickness (4)
Fig: Phase of transmission coefficient depending on (a) relative dielectric
constant (b) dielectric thickness.
Displacement Measurement
The phase difference between the reference and measured paths, produced by displacement of
the target location, is determined from in-phase (I) and quadrature (Q):
vI (t )  AI sin (t )
vQ (t )  AQ cos (t )
Where, (t) represents phase difference and can be determined by for an ideal quadrature
mixer:
 vI (t ) AQ
 (t )  tan 
 v (t ) A
I
 Q
1



Practically, quadrature mixers, however, have a nonlinear phase response due to their phase
and amplitude imbalances as well as DC offset. A more realistic from of the phase including the
nonlinearity effect can be expressed as:
 1

vI (t )  VOSI
A
 (t )  tan 
 tan  
 cos  ( A  A) v (t )  V

Q
OSQ


1
Displacement Measurement
The detected phase is generated by the time delay, , due to round trip-trip traveling of
electromagnetic wave for the distance between antenna aperture and target. Therefore, it has
relationship with range, r as:
4 f 0 r
c
where, f0 and c are operating frequency and speed in free space of electromagnetic
waves. The range as a function of time variable can be defined as :
 ( )  2 f 0 
r (t ) 
 (t )
0
4
Range variation is produced by changes in target location and can be expressed in
the time domain as:
r (nT )  r[nT ]  r[(n  1)T ],
n  1, 2,3,.....
where, T is the sampling interval. The displacement for the entire target
measurement sequence can be described as a summation of consecutive range
variations:
k
d (nT )   r (nT ),
n 1
n  1, 2,3,......., k
An Interferometric Sensor*
Fig: Measurement set-up for water level
gauging
Heterodyne Interferometric sensor. The target sits either on the XYZ axis (for displacement sensing) or on the
conveyor (for velocity measurement). The Reference channel is not needed for velocity measurement.
* Courtesy: Kim and Nguyen, IEEE Trans on Microwave Theory and Techniques, vol.52, p2503, Nov 2004
The block diagram of a Ka-band heterodyne measuring setup for lossy dielectric
materials.
Conclusions
Reflectometry is a non destructive sensors
With little modifications it is used in different applications