SIMULATION AND ANALYSIS OF SIMPLE ELECTROMAGNETIC
STRUCTURES WITH PRACTICAL APPLICATIONS AND PRESENTATION
OF ELECTROMAGNETIC CONCEPTS
A Project
Presented to the faculty of the Department of Electrical and Electronic Engineering
California State University, Sacramento
Submitted in partial satisfaction of
the requirements for the degree of
MASTER OF SCIENCE
in
Electrical and Electronic Engineering
by
Laurence Allen Breen. P.E.
FALL
2014
SIMULATION AND ANALYSIS OF SIMPLE ELECTROMAGNETIC
STRUCTURES WITH PRACTICAL APPLICATIONS AND PRESENTATION
OF ELECTROMAGNETIC CONCEPTS
A Project
by
Laurence Allen Breen, P.E.
Approved by:
__________________________________, Committee Chair
Milica Markovic, Ph.D.
____________________________
Date
ii
Student: Laurence Allen Breen
I certify that this student has met the requirements for format contained in the University format
manual, and that this project is suitable for shelving in the Library and credit is to be awarded for
the project.
__________________________, Graduate Coordinator ___________________
Preetham Kumar, Ph.D.
Date
Department of Electrical and Electronic Engineering
iii
Abstract
of
SIMULATION AND ANALYSIS OF SIMPLE ELECTROMAGNETIC
STRUCTURES WITH PRACTICAL APPLICATIONS AND PRESENTATION
OF ELECTROMAGNETIC CONCEPTS
by
Laurence Allen Breen, P.E.
In this project simulations of simple electromagnetic structures have been performed.
Inductive loop sensors and a ball grid array structures have been analyzed using 3D and 2D fullwave electromagnetic simulators along with scientific calculation software. Fundamental
electromagnetic concepts are presented and electromagnetic structures are simulated and
analyzed. Applications of these structures are presented. The characteristics of these applications
are compared to published data.
_______________________, Committee Chair
Milica Markovic, Ph.D.
_______________________
Date
iv
ACKNOWLEDGEMENTS
He who knows and knows not that he knows: He is asleep, wake him.
He who knows not and knows not that he knows not: He is a fool, shun him.
He who knows not and knows that he knows not: He is a child, teach him.
He who knows and knows that he knows: He is wise, follow him.
- Persian Proverb
In memory of Sean Hamilton McComas
GDX BDX
v
TABLE OF CONTENTS
Page
Acknowledgements.........................................................................................................................v
List of Tables............................................................................................................................ix
List of Figures...........................................................................................................................x
Chapter
1. INTRODUCTION.....................................................................................................................1
2. REVIEW OF FUNDAMENTAL CONCEPTS.........................................................................2
2.1
DC Conductor and Dielectric Losses.............................................................................2
2.2
Skin Effect.....................................................................................................................3
2.2.1 Skin Depth for an Infinite Half Plane..................................................................3
2.2.2 Infinite Half Plane...............................................................................................4
2.2.3 Skin Effect of a Cylindrical Conductor................................................................6
2.2.4 Internal Magnetic Flux of a Cylindrical Conductor.............................................7
2.2.5 Internal Impedance and Surface Resistance.......................................................10
2.2.6 H and Js for an Infinite Half Plane....................................................................12
2.3
Impedance as a Function of Frequency........................................................................13
2.3.1 Inductance..........................................................................................................16
2.3.2 Surface Roughness Effects................................................................................19
2.3.3 Capacitance........................................................................................................19
2.3.4 AC Dielectric Losses.........................................................................................21
2.4
Transmission Line Theory...........................................................................................21
2.4.1 Transmission Line Losses..................................................................................24
vi
2.4.2 Transmission Line Distortion............................................................................24
2.5
Classification of Partial Differential Equations...........................................................25
2.6
Effective Dielectric Constant...... ................................................................................27
2.7
Parameter Extraction....................................................................................................27
2.8
Overview of Chapter 2.................................................................................................30
3. SIMULATION OF INDUCTANCE OF SIMPLE ELECTROMAGNETIC STRUCTURES. 31
3.1
Partial Self-Inductance of a Cylindrical Conductor.....................................................31
3.1.1 2D Extractor Simulation....................................................................................33
3.1.2 Approximate Formula and Q3D Extractor Simulation......................................34
3.1.3 Results...............................................................................................................35
3.1.4 Analysis.............................................................................................................36
3.1.5 Partial Self-Inductance for a Varying Radius or Length....................................36
3.1.6 Result: Optimetrics, Variable Radius................................................................37
3.1.7 Analysis.............................................................................................................39
3.2
Partial Mutual Inductance of Two Conductors............................................................40
3.2.1 Simulation of Partial Mutual Inductance...........................................................40
3.2.2 Partial Mutual Inductance for Varying Pitch or Radius.....................................43
3.2.3 Analysis.............................................................................................................43
3.3
Inductance of a Rectangular Loop...............................................................................45
3.4
Simulations..................................................................................................................46
3.4.1 Single Value of Loop Dimensions.....................................................................46
3.4.2 Constant Loop Width With Varying Length......................................................46
3.4.3 Constant Loop Length With Varying Width......................................................49
vii
3.4.4 Constant Loop Area Held Constant With Varying Length and Width...............50
3.4.5 Analytical Formula............................................................................................51
3.5
Overview of Chapter 3..................................................................................................52
4. APPLICATIONS.....................................................................................................................53
4.1
Application to Inductive Loop Sensors........................................................................53
4.1.1 Theory...............................................................................................................53
4.2
Simulate Two Loops....................................................................................................55
4.3
Application to Ball Grid Arrays...................................................................................65
4.4
Simulate Parallel Plate With Vias Only and Parallel Plate With Field of Holes..........68
4.4.1 Analysis.............................................................................................................71
4.5
Overview of Chapter 4.................................................................................................71
5. CONCLUSIONS.....................................................................................................................72
Appendix.................................................................................................................................73
.
References...............................................................................................................................78
viii
LIST OF TABLES
Tables
Page
1.
Partial Self-Inductance for Varying Radius.....................................................................37
2.
Partial Self-Inductance for Varying Length.....................................................................38
3.
Calculated Partial Mutual Inductance Using Bogatin's Formula......................................43
4.
Simulated Partial Mutual Inductance Using Q3D-3D......................................................43
5.
Calculated Partial Mutual Inductance Using Bogatin's Formula......................................44
6.
Simulated Partial Mutual Inductance Using Q3D-3D......................................................44
7.
Comparison of Calculated and Simulated Values............................................................46
8.
Comparison of Calculated Mutual and Simulated Inductance.........................................48
9.
Comparison of Simulated and Calculated Loop Self-Inductance.....................................51
10.
Loop Dimensions.............................................................................................................57
11.
Single Loop Results.........................................................................................................57
12.
Two Loop Result.............................................................................................................59
13.
Single Loop With Metal Plate..........................................................................................61
14.
DC Results for Two Loops With One Shorted.................................................................62
15.
DC Resistance and DC Inductance for a Plane and Field of Holes..................................70
ix
LIST OF FIGURES
Figures
Page
1.
Infinite Half Plane.............................................................................................................4
2.
Skin Effect in Circular Conductor....................................................................................7
3.
Cross Section of an Infinite Torus....................................................................................9
4.
Section of an Infinite Torus. ............................................................................................9
5.
Crossover Frequency for a Round Conductor, Simulated With MATLAB....................15
6.
Types of Inductance.........................................................................................................17
7.
Lumped Equivalent Circuit..............................................................................................22
8.
PI Model of Two Port Admittance Network....................................................................28
9.
Reciprocal Model of Two Port Admittance Network......................................................28
10.
TEE Model of Two Port Impedance Network..................................................................29
11.
Single Cylindrical Conductor............................................................................................32
12.
Comparison of Q3D Extractor and Calculated Values of Partial Self-Inductance............35
13.
The Partial Self-Inductance for a Conductor of Varying Radius......................................38
14.
The Partial Self-Inductance for a Varying Length Conductor...........................................39
15.
Two Cylindrical Conductors.............................................................................................41
16.
Rectangular Loop..............................................................................................................46
17.
Constant Loop Width With Varying Length.....................................................................47
18.
Constant Loop Length With Varying Width.....................................................................49
19.
Inductance for Variable Length and Constant Area..........................................................50
20.
Two Loops.........................................................................................................................56
21.
AC Resistance and Inductance for a Single Loop.
x
Q3D.................................................58
22.
Q of a Single Loop.
Q3D.............................................................................................58
23.
AC Resistance and Inductance for Two Loops..................................................................59
24.
Mutual Inductance(Dashed Line) and Self Inductance(Solid Line) for Two Loops..........60
25.
Q of Two Loops. Q3D.....................................................................................................60
26.
Resistance and Inductance of Single Loop and Conducting Plate. ....................................61
27.
Q for Single Loop With a Conducting Plate......................................................................62
28.
AC Resistance and Inductance for Two Loops With One Shorted. Q3D...........................63
29.
Q for Two Loops With one Shorted. Q3D.......................................................................63
30.
Current Density.................................................................................................................64
31.
Electric Field.....................................................................................................................65
32.
Parallel Plate Transmission Line......................................................................................66
33.
Front, Showing Input and Output Vias.............................................................................66
34.
Back, Showing Blind Via Connecting Conductors...........................................................66
35.
Field Overlay Showing Surface Current Density, J A/m..................................................67
36.
Detail Field Overlay Showing Surface Current Density...................................................68
37.
Solid Parallel Plate Current Loop.....................................................................................69
38.
Parallel Plate With a Field of Holes..................................................................................69
39.
Sweep of AC R and L for Solid Parallel Plate. Simulated Using Q3D Extractor.............70
40.
Sweep of AC R and L for Field of Holes. Simulated Using Q3D Extractor.....................71
xi
1
Chapter 1
INTRODUCTION
1 INTRODUCTION
Computer simulation of electromagnetic structures is important in education, scientific and
commercial applications. Simulation of electromagnetic structures allows the student to
experiment and visualize electromagnetic principles. In scientific uses one can hypothesize new
concepts quickly and efficiently. In industry, prototyping can be done in software rather than
building the actual component. This saves time and ensures a better quality product.
Chapter 2 is a review of fundamental electromagnetic concepts that are related to simulation
and analysis of simple electromagnetic structures. In chapter 3 the simulation and analysis of
simple electromagnetic structures is described. Chapter 4 demonstrates applications of the
structures that have been simulated.
2
Chapter 2
REVIEW OF FUNDAMENTAL CONCEPTS
2 REVIEW OF FUNDAMENTAL CONCEPTS
In this chapter fundamental concepts needed to understand the simulation and application of
electromagnetic structures is presented. Section 2.1 examines DC losses in electromagnetic
structures. Section 2.2 examines the skin effect. Section 2.3 explains impedance in relation to
frequency. Section 2.4 is a brief review of transmission line theory. In section 2.5 partial
differential equations that are used in electromagnetic theory are reviewed. Section 2.6 a brief
explanation of the effect of a wave propagating through two dielectric materials. Section 2.7
shows the extraction of parameters from S-parameters using MATLAB calculations.
2.1 DC Conductor and Dielectric Losses
The DC loss is determined by the resistivity and the area of the of the conductor [10]
R
R =
L
Wt
ohm
(1)
DC resistance, ohm
 = Resistivity, ohm-m
L =
Length, W the width and t the thickness, m
The dielectric material is not a perfect insulator. The resistance of the dielectric is usually so
great that it is presumed to be infinite.
3
2.2 Skin Effect
Skin effect describes the phenomenon where high-frequency current tends to flow in a shell
around the inside circumference of a conductor[8]. When DC is applied to a conductor, the
current is distributed evenly across the cross-section of the conductor. The alternating current
induces a changing magnetic field in the conductor. This changing magnetic field induces a
changing electric field which opposes the change in the current at the center of the conductor.
2.2.1 Skin Depth for an Infinite Half Plane
Starting with Maxwell's equations, in phasor form [10]:
  E   j H
(2)
  H   j E   E
(3)
Using a vector identity for E, we get:
 

 2 E  j 2 2  1  j
E
 

(4)
The complex propagation constant, , for a general lossy medium is defined as
    j  j  1  j


(5)
The real part of Equation 5 is the attenuation constant, with units Np/m and the imaginary
part is the phase constant, with units rad/m.
4
In the case of a good conductor, (), then  can be approximated as:
    j  j 


 1  j 
j
2
(6)
Skin depth is the reciprocal of the attenuation constant. Separating the real term,

1
2



(7)
(m)
The skin depth, , is the thickness of the conductor which attenuates the fields by 1/e which
is 1 neper [8].
y
E 0, J 0
H0

x=
z
Jz
Jz
x
Figure 1: Infinite Half Plane.
2.2.2 Infinite Half Plane
Assuming a cosinusoidal signal and a surface current density of J 0 A/m, the current density
at some distance, x, into the half plane is given by Equation 8 [8].
5
J z ( x)  J 0 e
x

x

cos t  


JZ(x) =
Current density at a distance, x, into the conductor, A/m
J0
=
Surface current density (i.e. x=0), A/m

=
Skin depth, m

=
Angular frequency of the signal, radian/s
(8)
When this current density is integrated as a function of depth, the total current from the
surface to the depth, x, is given by Equation 9.
I z ( x)   J 0
  x 
x
x 


e  cos t    j sin  t   
2

 



IZ(x)
=
Current from surface to the depth, x, A/m
J0
=
The surface current density, A/m

=
Skin depth, m

=
Angular frequency of the signal, radian/s
(9)
Note that the current, IZ(x) is A/m, not A. Unless the width, y, is specified the y distance
would be infinite; the resulting current would be infinite.
The term, e– x/ expresses the decay of current density as 1 neper (1/e) at one skin depth.
When the right term is written in phasor form, it is clear that the phase will shift one radian at a
depth of one skin depth.
e
x

 j  t  



e
j
x

(10)
6
Equation 9 shows that the cosine term represents the real current flow and depending upon
its sign, current is in the positive or negative direction. When the sine term is positive it indicates
inductive current, (i.e., positive phase shift, with reference to J 0 ). When the sine term is negative,
it indicates capacitive current flow[8]. Real and reactive currents overlap. The forward and
reverse currents are separate as are inductive and capacitive currents. Since the conductor is
"good," there cannot be significant displacement current.
2.2.3 Skin Effect of a Cylindrical Conductor
Due to the skin effect, a cylindrical conductor will have its current diminish in the center
and increase at the outer radius of the conductor.
The volume current density, J, causes a magnetic field, H, to result.
 
 H  J
(11)


B
 E  
t
(12)

  B
 H  J 
t
(13)
7
J
J
H
E
H
E
H
Figure 2: Skin Effect in Circular Conductor.
2.2.4 Internal Magnetic Flux of a Cylindrical Conductor
The inductance of a conductor is its self inductance. Self-inductance is composed of an
internal and an external inductance. Internal inductance is due to the flux linkages inside the
conductor. External inductance is due to the flux linkages outside of the conductor [4].
8
Figure 3 is the cross section of an infinite length conducting torus. Figure 4 is a short
section of the infinite torus. The torus has a uniform, low frequency current, I. There is a
cylindrical shell of radius, , and differential thickness of d. The magnetic field is indicated by
the counter-clockwise circling lines within the radius of the shell.
The magnetic flux within the cylindrical shell in Figure 3 is given by Equation 14. At low
frequency, the magnetic field is uniform throughout the conductor. The term in the integral
expresses the area of the shell proportional to the area of the entire cross section in Figure 3 [4].
  2
m  
d
  a2

(14)
The right hand term in the integral, Equation 15 is the magnetic flux density, B, through the
area of region 2 in Figure 4. Integral Equation 15 calculates the magnetic flux through region 1 in
Figure 4. Region 3 is the differential area, h · d.
a
    2   I h
Ih
m   
d


   a 2  2a 2
8
0 

m
=
Magnetic flux through region 1, Wb
The result of Equation 15 shows that the magnetic flux (Wb) for a cylindrical conductor
depends only upon amount of current and the length of the segment.
(15)
9
a
B
d
d

Figure 3: Cross Section of an Infinite Torus.
I

h
I
a
1 23
Figure 4: Section of an Infinite Torus.
10
2.2.5 Internal Impedance and Surface Resistance
Since fields decay rapidly over distance in a conductor the fields may be considered to
penetrate only one skin depth. Surface approximations, such as surface current and surface
impedance are evaluated over one skin depth.
For lossless media, the intrinsic impedance and the wavenumber (propagation constant) is
[10].




(16)
1
m
k   
For lossless media, where 0 is the value of  in Equation (5),the propagation constant is
expressed by Equation 17 [10].
  j 
1
m
(17)
A general lossy medium has a wave impedance

j


(18)
Internal impedance is defined as Equation 19 [11].
Zi 
E0
I0
(19)
Zi
=
Internal impedance, /m
E0
=
Electric field at z=0, V/m
I0
=
Current at z=0, A
11
The internal or surface impedance is calculated using Equation 20 [16].
Z s  1  j 
 l

 w
j
l
w
(20)
ZS
=
Internal impedance, 
l,w
=
Length, width, m
The resistance, R, and reactance, X, are calculated using Equation 21.
Z s  R  jX  1  j   f  
l
w
(21)
Note that the resistance and reactance are equal for a good conductor.
Surface resistivity, for a square medium with a thickness of one skin depth is calculated
with Equation 22.
Rs 
Rs

  f

=
(22)
Surface resistivity, /square
The surface resistance for a conductor with a specified length and width and a thickness of
one skin depth is:
R  Rs
l
w
(23)
R
=
Surface resistance, 
w
=
Width, along y-axis, m
l
=
Length, along z-axis, m
12
The internal reactance, X, Equation 21 is used to find the internal inductance.
L
L
X
l 


w 2
=
(24)
Internal inductance, H
The average depth of the current is half of the thickness of the skin depth.
2.2.6 H and Js for an Infinite Half Plane
Figure 1 depicts an infinite half plane. The y-z plane is a boundary between air and a "good"
conductor (  ) [14]. An electromagnetic plane wave impinges from the air above the y-z
plane. Because of the boundary condition the tangential E-field is continuous across the
boundary, so a plane wave propagates through the conductor in the x direction. The E field
causes a current to flow in the z direction, having a current density of J z. The magnitude of the
current density decays exponentially over depth in the x direction [11].
d 2 EZ
 j   EZ
dx 2
(25)
The solution of Equation 25 is:
E Z  E0
x  j x
e e 
(26)
E0 is the electric field intensity (V/m) at the surface (x=0) of the infinite half plane, as
shown in Figure 1.
Similarly, Hy and Jz can be derived:
HY  H 0
x  j x
e e 
(27)
13
H0 is the magnetic field intensity (A/m) at the surface.
JZ  J0
x  j x
e e 
(28)
J0 (=  E0) is the surface current density (A/m).
2.3 Impedance as a Function of Frequency
Low frequency resistance of a cylindrical conductor is approximated by [6]:
R DC 

a
RDC
=

Resistivity, ohm-m
=
a =
(29)
Low frequency resistance, /m
Cross sectional area, m2
High frequency resistance of a round conductor is given by:
ReR AC  
1
p 
RAC
=
High frequency resistance, /m
p
=
Perimeter of the conductor, m

=
Skin depth, m

=
Conductivity, S/m
(30)
The skin effect onset frequency,  Equation 31, also called the crossover frequency, is the
frequency at which low frequency (Equation 29) and high frequency (Equation 30)
approximations of resistance are equal.
14
 
2  p
 
  a 
2
(31)

=
Crossover frequency, rad/s

=
Permeability, H/m

=
Conductivity, S/m
p
=
Perimeter of the conductor, m
Equation 32 is an accurate formula for calculating the internal impedance of a round
conductor. Figure 5 is a graph of zi Equation 32.
zi 


 I 0 r j 
2r I 1 r j 


zi
=
Internal impedance for a round conductor, ohm/m

=
Intrinsic impedance, ohm
I0 , I1 =
Bessel functions

=
Permeability, H/m

=
Conductivity, S/m

=
Frequency, rad/s
(32)
Im p e d a n c e , o h m /m
15
10
2
10
1
10
0
10
-1
10
6
10
7
10
8
10
9
F re q u e n c y, H z
Figure 5: Crossover Frequency for a Round Conductor, Simulated With MATLAB.
In Figure 5 the solid curve is the real part of the internal impedance and the dashed curve is
the imaginary part of the impedance. The real part of the internal impedance is constant up to the
crossover frequency then increases at 10 dB per decade. Inductance is constant up to the
crossover frequency then decreases at 10 dB per decade. Due to the skin effect, the resistance
increases and the internal inductance decreases. At frequencies higher than the crossover
frequency the real and imaginary parts of the impedance both increase at 10 dB per decade.
The internal impedance below the crossover frequency for a round conductor is
approximated using Equation 33.
When  << 
zi 
1

 j
a
8
(33)
16
The internal impedance above the crossover frequency is approximated using Equation 34.
When  >> 
zi 

p 
e
j
4
(34)
zi
=
internal impedance, ohm/m

=
Conductivity, S/m
a
=
Area, m2

=
Frequency, rad/s

=
Permeability, H/m
p
=
Perimeter of the conductor, m
Figure 5 shows the MATLAB simulation of the crossover frequency.
z
R ac 2  R dc 2
z
=
Internal impedance, /m
RAC, RDC
=
High and low frequency, /m
(35)
2.3.1 Inductance
There are several ways that inductance can be defined.
A 1 henry (H) inductance is defined as a 1 V emf resulting from a 1 ampere, 1 Hz current in
a conductor [5].
17
emf   L
I
t
emf
=
Electromotive force, V
L
=
Inductance, H
I
=
Current, A
t
=
Time, s
(36)
An alternative definition of inductance is the amount of magnetic flux resulting from a
current. A flux of 1 Wb resulting from a current of 1 A has an inductance of 1H.
Magnetic flux is defined as a 1 T magnetic flux density through an area of 1 m 2 [5]. This
results in a magnetic flux of 1 weber (Wb) [2].
L
N
I
(37)
L =
Inductance, Henry or Wb/A
 =
Magnetic flux, Wb
N =
Number of turns in the coil
I =
Current, A
Types of
Inductance
Self
Inductance
Mutual
Inductance
Internal
Inductance
External
Inductance
Figure 6: Types of Inductance.
18
Self-inductance the amount of magnetic flux generated by its own current of 1 ampere.
Mutual inductance is the amount of magnetic flux generated in the first conductor due to a
current in a second conductor.
Loop self-inductance: The magnetic flux surrounding a conductive loop.
Loop mutual inductance: The magnetic flux regarding two current carrying loops [2].
Partial Self-inductance: The magnetic flux surrounding a section of a conductor.
Partial Mutual self-inductance: The magnetic flux regarding two current carrying
conductor sections.
Effective inductance: The amount of magnetic flux surrounding a section of a loop that is
due to 1 ampere of current in the entire loop.
Equivalent inductance: The amount of self-inductance of one conductor resulting from a
collection of other conductors.
At DC or low frequency, where the skin effect is not significant, the current is uniformly
distributed throughout the cross section of the conductor [4].
As the frequency is increased, the skin effect causes the current to move to the outer part of
the conductor. External inductance is due to the current on the surface of the conductor [2].
External inductance can be determined by measuring the inductance at frequency that is high
enough to cause the skin effect to eliminate most of the internal inductance. Then the only
inductance will the external inductance.
19
2.3.2 Surface Roughness Effects
Due to the skin effect, at high frequencies, current is forced toward the surface of the
conductor. The surface of a conductor is not perfectly smooth. When the skin depth is
comparable to the depth of the roughness, the roughness will significantly increase the resistance
of the conductor.
Pozar gives a formula for the attenuation constant in the presence of surface roughness
Equation 38 [10].
2
 2
 

1
 c   c 1  tan 1.4  
   
 
c'
=
Attenuation considering surface roughness, Np/m
C
=
Attenuation for a smooth conductor, Np/m

=
RMS surface roughness, m

=
Skin depth, m
(38)
2.3.3 Capacitance
Capacitance is defined as Equation 39 [5].
C
Q
V
(39)
C
=
Capacitance, Farad
Q
=
Charge on one plate , C
V
=
Difference of potential between plates, V
For a 1C charge on one of the plates with a difference of potential of 1 V the capacitance is
1F.
20
Capacitance may also be defined as the energy stored in a capacitor Equation 41.
W
1
CV 2
2
W
C
=
(40)
Electric energy, J
2W
V
2
(41)
Equal and opposite charges that are separated are called an electric dipole. A dipole is
characterized by the dipole moment, given by Equation 42.
p  Qd
(42)
p
=
Dipole moment, C-m
Q
=
Charge, C
d
=
Distance between charges, m
The direction of the dipole is from the negative to the positive charge.
The polarization the sum of the individual electric dipole moments Equation 43.
P
1
p
V
(43)
P
=
Polarization, C/m2
p
=
Dipole moment, C-m
The displacement is given by Equation 44.
D E P
D
=
Displacement, C/m2
E
=
Electric field intensity, V/m
(44)
21
2.3.4 AC Dielectric Losses
The dielectric constant of a material is given by
   ' j ' '

=
Complex dielectric constant, F/m
'
=
0 r , which is the permittivity, F/m
'' =
(45)
Imaginary part, representing the losses, F/m
The loss tangent is the quotient of the imaginary part to the real part of the displacement
current.
Equivalently, it is the ratio of energy loss to stored energy.
tan  
   
 

=
Conductivity of the dielectric material, S/m

=
Angular frequency, r/s
(46)
The numerator of Equation 46 shows that dielectric loss is can not be distinguished from
conductivity loss [10].
2.4 Transmission Line Theory
At frequencies where the length of the transmission line smaller than 1/10 of a wavelength,
the transmission line can be approximated using Kirchoff's voltage and current laws. A signal in
a transmission line will reflect back and forth (unless the input and output impedance is matched)
eventually settling to a steady state. The time for the signal to settle is shorter than the period of
22
the signal for a low frequency signal. A high frequency signal will have a period comparable to
the settling time of the transmission line.
The lumped equivalent circuit of a transmission line is shown in Figure 7 [4].
i(x,t)
L Dx
i(x+Dx,t)
R Dx
G Dx
v(x,t)
C Dx
v(x+Dx,t)
Figure 7: Lumped Equivalent Circuit.
Using Kirchoff's current law, in the time domain, we have:
i ( x, t )  G x v ( x  x, t )  C x
 (v( x  x, t ))
 i ( x  x, t )  0
t
(47)
Using Kirchoff's voltage law:
v ( x, t )  R x i ( x, t )  L x
 (i ( x, t ))
 v( x  x, t )  0
t
(48)
When Equation 47 and Equation 48 are divided by x and the limit is taken as x → 0,
resulting in the time-domain form:
 v( x, t )
i ( x, t )
  R i ( x, t )  L
x
t
(49)
 i ( x, t )
v ( x, t )
 G v ( x, t )  C
x
t
(50)
Equation Equation 49 and Equation 50 are the telegrapher's equations .
23
In phasor notation we have:
dV
 ( R  jL) I   Z I
dx
(51)
Where Z is the series impedance (ohm/meter) [9].
dI
 (G  jC ) V  Y V
dx
(52)
Where Y is the shunt admittance (siemens/meter).
Equations Equation 51 and Equation 52 are solved simultaneously, resulting in the wave
equations for V and I.
d 2I
dx
2
d 2V
dx
2
  2I  0
(53)
  2V  0
(54)
where,
    j  ( R  jL)(G  jC )
(55)
is the complex propagation constant, expressing the loss () and phase shift per unit length
() of the transmission line.
The linear, second-order differential equations Equation 53 and Equation 54 can be solved,
resulting in:
V  V0 e  x  V0 e  x
(56)
I  I 0 e  x  I 0 e  x
(57)
24
Dividing Equation 51 by Equation 52, we get Z0, the characteristic impedance.
Z0 
R  jL
G  jC
(58)
2.4.1 Transmission Line Losses
A transmission line's parameters, capacitance, inductance, resistance and conductance are
used to define the characteristics of fairly simple lines [10]. The CLRG is useful for an intuitive
view of transmission lines. Capacitance models the electric field between the conductors.
Inductance models magnetic field resulting from the displacement and conduction currents.
Resistance models the resistive loss from the conductors. Conductivity models the dielectric loss.
The total transmission line loss is the sum of the individual losses [15].
  c  d  r  l
c
=
conductor losses, Np/m
d
=
dielectric losses, Np/m
r
=
radiation losses, Np/m
l
=
leakage losses, Np/m
(59)
2.4.2 Transmission Line Distortion
The phase constant, , is a function of frequency. This means that portions of the signal that
are at differing frequencies will propagate at different velocities. The signal will spread, causing
distortion in the transmitted signal. This spreading is called dispersion [10].
25
Frequency dispersion is where a transmission line has a different group velocity at different
frequencies. This applies to non-TEM transmission lines.
2.5 Classification of Partial Differential Equations.
Electromagnetic problems use second-order, linear differential equations [12].
2
2
2


L  a 2 b
c 2 d e  f
x
xy
y
x
y
(60)
Equation Equation 60 is a differential operator. It can be classified by the a,b and c
coefficients as [3]:
elliptic when b 2  4ac  0
parabolic
 0
hyperbolic
 0
Elliptic PDE's (partial differential equations) are used for steady-state problems for
Laplace's equation.
 2T  2T

0
x 2 y 2
(61)
Parabolic PDE's are used for problems such as the diffusion of heat.
 2T
T
k
2
x
t
(62)
26
Hyperbolic PDE's are the wave equation.
2 y 1 2 y

x 2 c 2 t 2
(63)
Laplace's equation expresses Gauss' law. Generally speaking, Gauss' law says that a region
with no charge has no total flux [1]. Laplace's equation is also interpreted as potential at some
point is the average the surrounding potentials.
Keeping in mind that the second derivative expresses the sense of concavity of a function,
the concavity of the two differential equation are equal and opposite. This is seen in the plot of
the electric field and the potential curves. The curves are perpendicular at all locations and have
the same, but opposite, curvature.
The Laplace equation Equation 64, a second order, partial differential equation, is derived
from the Maxwell equations.
2V 2V

0
 x2  y 2
(64)
 V  E
(65)
2 V  0
(66)
27
2.6 Effective Dielectric Constant
The effective dielectric constant ( eff) is the result of a wave propagating through different
dielectrics, such as fiberglass and air as in a microstrip, for example. The effective dielectric
constant will be different from that of either of the dielectrics.
The effective dielectric constant is stable at high or low frequencies but will vary in
between. At higher frequency, more of the wave propagates through the dielectric material than
through the air; the effective dielectric constant increases with frequency.
2.7 Parameter Extraction
The transmission line is simulated and the S parameters extracted. The extracted
S parameters are converted to ABCD parameters using conversion tables in [10]. The ABCD
parameters are then used to find the admittances for Figure 8.
For a Pi network
  

Ln   imag
 Y n  
(67)
 Y n  
C n   imag

  
(68)
L
=
Inductance of individual item in two-port network
C
=
Capacitance of individual item in two-port network
n
=
Identifying number individual item in two-port network
28
Y3
Y1
Y2
Figure 8: PI Model of Two Port Admittance Network
Due to reciprocity, Y12 = Y21. The Pi model, that uses admittance parameters, is shown in
Figure 9.
-Y12
Y11+Y12
Y22+Y12
Figure 9: Reciprocal Model of Two Port Admittance Network
29
Z1
Z2
Z3
Figure 10: TEE Model of Two Port Impedance Network
A MATLAB '.m' file calculates the values for Z1, Z2 and Z3 for a lossless network. The
inductance and capacitance of each item is calculated. Its input is the S-parameters (s2p file) of
the two port network that will be converted to a TEE network. The MATLAB program uses
Equation 69 and Equation 70 to calculate the values of the components.
A copy of the '.m' file (TWO_PORT_TEE.m) is in the appendix at the end.
For a Tee network
 Z n  
Ln   imag

  
(69)
  

C n   imag
 Z n  
(70)
L
=
Inductance of individual item in two-port network
C
=
Capacitance of individual item in two-port network
n
=
Number identifying the individual items in two-port network
30
The MATLAB '.m' file (TWO_PORT_PI) calculates the values for Y1, Y2 and Y3 for a
lossless network. Its input is the S-parameters of the two port network that is to be converted to a
PI network. The MATLAB program uses Equation 71 and Equation 72 to calculate the values of
the components.
A copy of the '.m' file is in the appendix at the end.
For a Pi network
  

Ln   imag
 Y n  
(710
 Y n  
C n   imag

  
(72)
2.8 Overview of Chapter 2
Chapter 2 shows the theory needed to understand the electromagnetic concepts.
31
Chapter 3
SIMULATION OF INDUCTANCE OF SIMPLE ELECTROMAGNETIC STRUCTURES
3 SIMULATION OF INDUCTANCE OF SIMPLE ELECTROMAGNETIC STRUCTURES
In this chapter electromagnetic structures are simulated and the results are compared to
values given by calculation of analytic equations. Section 3.1 shows the partial self-inductance of
a cylindrical conductor. Section 3.2 demonstrates the partial mutual inductance of two
conductors. In Section 3.3 the inductance of a rectangular loop is shown. Section 3.4 several
structures are simulated.
3.1 Partial Self-Inductance of a Cylindrical Conductor
In this section the partial self-inductance for a single cylindrical conductor is demonstrated.
The structure was simulated and compared to the results of the values calculated using a closedform approximation formula Equation 73 [2].
 
3

L  5d ln 2d  
4
 r
L =
partial self-inductance, nH
r =
radius of wire, inch
d =
length of wire, inch
(73)
32
Figure 11 shows a single conducting rod of radius, r, and length d.
Figure 11: Single Cylindrical Conductor.
The simulations of Q3D Extractor have assumptions of the range of DC and AC
calculation, using Equation 74 and Equation 75.
When the signal frequency is lower than the upper DC frequency, Equation 74 Q3D
Extractor assumes that the skin depth is greater than the thickness of the conductor.
upper dc freq 
tcond
=
1
    tcond 2 0
(74)
Thickness of the conductor, m
When the signal frequency is greater than the lower ac frequency, Equation 75 Q3D
Extractor assumes that the skin depth is much smaller than conductor thickness. It is presumed
that all of the current is on the surface of the conductor and that the internal inductance has
reached its minimum. Further increases in signal frequency are presumed to no longer affect the
skin depth.
lower ac freq 
9
    tcond 2  0
(75)
33
Between the upper DC and lower AC frequencies, Q3D Extractor uses an equivalent circuit
model to approximate the resistance and inductance.
The DC resistance of a cylindrical conductor can be calculated by assuming that the current
flows through the whole cross sectional area of the conductor Equation 76.
dc resistance 
length
area  
(76)
The AC resistance of a cylindrical conductor can be estimated by calculating the conducting
area assuming that no current flows closer to the center of the conductor than the skin depth
Equation 73 [10] .
ac resistance 
length
 s  circumfrence  
length
=
length of conductor, m
s
=
skin depth, m
circumference =

=
(77)
circumference of conductor, m
conductivity of copper, S/m
3.1.1 2D Extractor Simulation
The program that is used to do simulations is Q3D Extractor, which is composed of both a
two- and a three-dimensional simulator and the Optimetrics simulator. The name, "Q3D
Extractor," refers to the whole program and to the three-dimensional simulator component of the
package.
34
A three-dimensional simulation is done using Q3D Extractor. A two-dimensional simulation
is done using 2D Extractor. The Optimetrics simulator simulates a model by varying a parameter.
For example the geometric dimensions or the relative permittivity can be swept.
The single conductor cannot be simulated using 2D Extractor. A 2D Extractor simulation
requires a reference ground. In this case there is only a single conductor. Putting in a reference
ground (e.g. a ground plane) would alter the field around the conductor, changing the partial selfinductance. The partial self-inductance, for a fixed frequency, radius and length, is calculated
and the result is compared to the Q3D Extractor simulation.
3.1.2 Approximate Formula and Q3D Extractor Simulation
The partial self-inductance is calculated using Equation 73 [2]. It is not known if the
equation is for DC or AC. The DC resistance is calculated using Equation 76. The AC resistance
is calculated using Equation 77.
The partial self-inductance is simulated using Q3D Extractor. An simulation is done for a
rod of radius 0.254 mm and a length of 2.54 mm. The material is copper. The inductance is
simulated using the Q3D Extractor. The resistance and inductance are simulated at DC and at 1
GHz.
The source and sink are at opposite ends of the rod. The voltage reference is at infinity. In
Q3D Extractor the charge of the model is balanced by a charge at infinity. The simulation is done
at 1 GHz, which ensures that the model is well into the skin effect region (this was spot-checked
at 10 GHz). The model is solved for the DC and the AC resistance and inductance using Q3D
Extractor solver.
35
3.1.3 Results
Figure 12 shows the comparison of the calculated value of the partial self-inductance to the
results of the Q3D Extractor simulation. The AC resistance Equation 77 and DC resistance
Equation 76 are also shown, but not included in the discussion.
Calculated Results
Inductance calculated using (73)
1.1229 nH
RDC = .21558 m RAC = 13.12 m
Q3D Extractor Results
Q3D Extractor, DC:
Resistance
Inductance
0.21856 m
1.1873 nH
Q3D Extractor, AC at 1 GHz:
Resistance
Inductance
13.17 m
1.0796 nH
Figure 12: Comparison of Q3D Extractor and
Calculated Values of Partial Self-Inductance.
36
3.1.4 Analysis
Referring to Fig. 12, the calculated inductance Equation 73 is about halfway between the
AC and DC simulated values. The difference between the approximated and simulated values of
DC inductance is about 6 % and between the AC values is about 4 %.
At DC the internal inductance is at its maximum. At 1 Ghz, the conductor has a skin depth
of 2.09 μm. The radius of the conductor is 254 μm. The conductor is well into the skin effect
region and there is little current inside the conductor. The difference of inductance between the
AC and DC simulated results is about 10 %. This result shows that the amount of internal
inductance is significant.
3.1.5 Partial Self-Inductance for a Varying Radius or Length
Two parametric sweeps are done using Optimetrics: partial self-inductance as a function of
the length, with the radius held constant and as a function of the radius of the conductor, while
holding the length constant. These parametric sweeps are done at 1 GHz. The length is swept
from 2.54 mm to 6.0 mm, while holding the radius constant at 0.254 mm. The radius is swept
from 0.254 mm to 0.508 mm while holding the length constant at 2.54 mm. Optimetrics sweeps
use the variables $r as the radius and $d as the length.
The resulting inductance values of the simulation and the calculation Equation 73 are
compared and graphed in Figure 13 and 14. The simulation is done using a Q3D Extractor
Optimetrics simulation.
37
3.1.6 Result: Optimetrics, Variable Radius
The partial self-inductance for a varying radius conductor, with a fixed rod length, is shown
in Table 1. The simulated and calculated values are compared. The data of Table 1 are plotted in
Figure 13. The solid line is the partial self-inductance calculated using Equation 73 [2]. The
dashed line is the result of the Q3D Extractor simulation.
Table 1: Partial Self-Inductance for Varying Radius
radius
mm
0.254
0.279
0.305
0.330
0.356
0.381
0.406
0.432
0.457
0.483
0.508
Vary Radius
Calculated
Simulated
nH
nH
1.123
1.080
1.075
1.036
1.032
0.998
0.992
0.963
0.955
0.931
0.920
0.901
0.888
0.874
0.858
0.849
0.829
0.826
0.802
0.804
0.776
0.783
Rod Length = 2.54 mm
% Difference
4.02
3.76
3.45
3.05
2.58
2.05
1.56
1.03
0.37
-0.25
-0.95
38
A C In d u c t a n c e , n H
1 .1 0
1 .0 5
1 .0 0
0 .9 5
0 .9 0
0 .8 5
0 .8 0
0 .7 5
0 .2 5
0 .3 0
0 .3 5
0 .4 0
R a d iu s , m m
0 .4 5
0 .5 0
Figure 13: The Partial Self-Inductance for a Conductor of Varying Radius.
The dashed line is the result of the Q3D Extractor simulation, the solid line is the result of
Equation 73 results of partial self-inductance and the solid line shows the calculated values.
The partial self-inductance for a varying length conductor, with a fixed radius, is shown in
Table 2. The simulated and calculated values are compared.
Length
mm
2.540
2.886
3.232
3.578
3.924
4.270
4.616
4.962
5.308
5.654
6.000
Vary Length
Calculated
Simulated
nH
nH
1.123
1.080
1.348
1.287
1.582
1.508
1.823
1.736
2.071
1.972
2.324
2.211
2.583
2.458
2.848
2.711
3.117
2.962
3.390
3.223
3.668
3.488
Rod Radius = 0.254 mm
% Difference
Table 2: Partial Self-Inductance for Varying Length.
4.02
4.72
4.87
4.99
5.00
5.10
5.09
5.07
5.22
5.19
5.16
39
The data of Table 2 are plotted in Fig. 14. The solid line is the partial self-inductance
calculated using Equation 73 [2]. The dashed line is the result of the Q3D Extractor Optimetrics
simulation.
Note that this plot is for a varying conductor length as is Figure 6-7 in the Bogatin [2] book.
In the text the vertical axis is logarithmic and the horizontal axis starts at a shorter length. This is
why the two graphs look quite different.
4 .0 0
A C In d u c ta n c e , n H
3 .5 0
3 .0 0
2 .5 0
2 .0 0
1 .5 0
1 .0 0
2 .5 0
3 .0 0
3 .5 0
4 .0 0
4 .5 0
L e n g th , m m
5 .0 0
5 .5 0
6 .0 0
Figure 14: The Partial Self-Inductance for a Varying Length Conductor.
3.1.7 Analysis
Optimizer is used to sweep the length and radius. The resistance is not simulated using
Optimizer; it is not needed and would slow the simulation.
The results of varying the radius of a conductor while holding the length constant are
compared in Table 1. The partial self-inductance calculation is done using Equation 73. The
simulation is done using Optimizer.
40
The percentage difference between the calculated and the parametric simulation of selfinductance is at most about 4%, with the calculated values being greater than the parametric
values.
The reason for this discrepancy is the approximate formula Equation 73 does not take
frequency into account. Presumably, the result is the DC partial self-inductance. The solution
from the Q3D Extractor DC simulation is done at 100 MHz and should have a lesser value than
the value calculated at DC.
The Q3D Extractor simulation was done at 1 GHz. The simulation frequency could be
"tweaked" so that the simulation results would more closely match the approximated result. But
if the simulation frequency were at the "DC approximation" frequency, the simulated inductance
values would be greater than the approximated values.
The simulation calculation was done with a longer conductor (1 inch or 25.4mm) and the
approximate calculated value was still about 4% greater than the Q3D Extractor value.
3.2 Partial Mutual Inductance of Two Conductors
Demonstrate the partial mutual inductance between two conductors. Simulate the structure
and verify the simulation using an approximation formula Equation 79 [ 2].
In the following section, the partial mutual inductance for two conductors of varying length
and radius is simulated and the results are compared to those of an approximation formula.
3.2.1 Simulation of Partial Mutual Inductance
Partial mutual inductance is given by Equation 78 [7]:
41
M
M=
21
i1
(78)
mutual inductance, Henry
21 = flux linking conductor both conductors due to the current in conductor 1, Weber
i1 =
current in conductor 1, Ampere
The partial mutual inductance for two cylindrical conductors is approximated using
Equation 79 [2].
2

s  s  
M  5d ln 2d  1     
d  2d  
 s

 
M=
mutual inductance, nH
d =
length of rods, inch
s =
distance between rod centers, inch
Figure 15 shows two conducting rods, separated at a distance, s, and of length, d.
Figure 15: Two Cylindrical Conductors.
(79)
42
The partial mutual inductance for the two rods cannot be simulated in 2D Extractor. The 2D
simulator requires that one of the conductors be a reference ground. The reference ground in the
2D simulator is not included in the solution matrix. If another conductor is were added to
function as the reference ground, the field around the rods would be altered. The resulting
simulation would have an incorrect value for the partial mutual inductance.
For this simulation, Q3D Extractor has two nets, one for each conductor. A net is one source
and sink, that is, a current input and an output. One is used for the signal while the other is used
as a reference. The reference conductor is the return path for the signal and is assigned a current
of -1 amp. This means that the conductors are in odd mode (currents in opposite directions). This
(even or odd mode) affects the partial mutual inductance but not the partial self-inductance.
Although they do not apply directly to partial mutual inductance, other simulations are done here.
In the simulation, the inductance is calculated at 100 MHz. Q3D's online help gives
formulas to estimate the lower frequency for ac and the upper frequency for dc calculations
Equation 74 Equation 75. The radius of the conductors is 0.027 mm and the pitch (center-tocenter separation) is 0.127 mm. Because these dimensions are small, the simulation runs quickly.
There is no frequency sweep.
The radius and the pitch of the conductors is expressed with the variables $rad and $pitch.
Variables are used because it is easy to change dimensions of the structure. Locations of the
components are made a function of input variables so that a change in one dimension causes the
other necessary changes in the dimensions. The variables can be viewed and edited using the
project variables dialog box.
43
3.2.2 Partial Mutual Inductance for Varying Pitch or Radius
Simulations for the two cylindrical conductors are done using Q3D Extractor. A parametric
simulation is run to demonstrate the change in partial mutual inductance as the pitch is varied.
The radius of the rods is kept at 0.027 mm and the pitch is varied 0.079 mm to 0.254 mm. The
result can be viewed as a graph or a table in the post simulation display. The approximation
formula Equation 79 does not have the radius of the rods as a variable.
Partial mutual inductance is simulated using Q3D Extractor. A simulation is run with rods
of 0.027 mm radius, 1 mm length and 0.127 mm pitch.
Skin effect greatly affects the AC resistance. The internal inductance is much less than the
external inductance, so the inductance is not greatly changed with frequency.
3.2.3 Analysis
Vary Rod Length
inch
mm
nH
0.0197 0.5000 0.1281
0.0236 0.6000 0.1708
0.0276 0.7000 0.2168
0.0315 0.8000 0.2655
0.0354 0.9000 0.3167
0.0394 1.0000 0.3700
0.0433 1.1000 0.4253
0.0472 1.2000 0.4824
0.0512 1.3000 0.5411
0.0551 1.4000 0.6013
0.0591 1.5000 0.6629
Separation = 0.127 mm
Table 3: Calculated Partial Mutual
Inductance Using Bogatin's Formula.
Vary Rod Length
mm
nH
0.5000
0.1314
0.6000
0.1746
0.7000
0.2208
0.8000
0.2704
0.9000
0.3226
1.0000
0.3762
1.1000
0.4325
1.2000
0.4908
1.3000
0.5507
1.4000
0.6113
1.5000
0.6746
Separation = 0.127 mm
Table 4: Simulated Partial Mutual
Inductance Using Q3D-3D.
44
Referring to Tables 3 and 4 it can be seen that as the rod length is varied, the partial mutual
inductance increases. This is because the longer the rods, the more flux linkages will be
available. The greatest discrepancy between the calculated and the simulated values is 2.5%.
Vary Separation
inch
mm
nH
0.0031 0.0790 0.4545
0.0038 0.0965 0.4184
0.0045 0.1140 0.3889
0.0052 0.1315 0.3640
0.0059 0.1490 0.3426
0.0066 0.1665 0.3239
0.0072 0.1840 0.3074
0.0079 0.2015 0.2926
0.0086 0.2190 0.2793
0.0093 0.2365 0.2672
0.0100 0.2540 0.2562
Rod Length = 1.0 mm
Vary Separation
mm
nH
0.0790
0.4636
0.0965
0.4259
0.1140
0.3956
0.1315
0.3701
0.1490
0.3482
0.1665
0.3299
0.1840
0.3130
0.2015
0.2977
0.2190
0.2841
0.2365
0.2721
0.2540
0.2612
Rod Length = 1.0 mm
Table 5: Calculated Partial Mutual
Inductance Using Bogatin's Formula.
Table 6: Simulated Partial Mutual
Inductance Using Q3D-3D.
Referring to Tables 5 and 6, it can be seen that as the separation of the rods is varied,
keeping the radius constant, that the partial mutual inductance decreases. This is because the
greater the separation, the lesser the intensity of the linking field. The discrepancy between the
calculated and the simulated values is 2%.
45
3.3 Inductance of a Rectangular Loop
It is often presumed that the loop inductance depends only upon the area inside of a loop.
The loop inductance is also dependent upon the proximity of the sides of the loop [2]. Given a
long rectangle and a square, both having the same area, the rectangle will have greater loop
inductance.
The loop self-inductance of a single turn, rectangular loop of rectangular wire is given by
Equation 80 [15]. This is the quasi-static inductance.


 2  s1  s 2 
  s1  logs1  g   s 2  logs 2  g 
Lloop  0.02339  s1  s 2 log
 bc 


s1  s 2


 0.01016  2 g 
 0.447(b  c )
2


g

s12  s 22
s1, s2 =
length and width of the loop, inch
b, c
width and thickness of conductor, inch
=
(80)
The DC resistance of the loop is calculated using Equation 81. This does not take skin effect
into account.
Rdc 
length

area
length =
length of conductor, m
area
=
cross sectional area of conductor, m2

=
resistivity of conductor, -m
(81)
46
3.4 Simulations
Figure 16: Rectangular Loop
3.4.1 Single Value of Loop Dimensions
The rectangular loop structure is simulated using Q3D extractor. Figure 16 shows the width
and length of the loop. The length of the loop is 5 mm, width is 2.5 mm for an area of 12.5 mm2 .
The cross section of the conductor is square, 0.05 mm2. The simulation frequency is 1 MHz. The
loop self-inductance is calculated using Equation 80 and the resistance for the loop using
Equation 81. The results are compared in Table 7.
Table 7: Comparison of Calculated and Simulated Values
Calculated
Q3D Extractor
Resistance
103.783 m
103.25 m
Loop Inductance
12.742 nH
12.92 nH
3.4.2 Constant Loop Width With Varying Length
The loop self-inductance is simulated as the length of the loop is varied using Q3D
Extractor Optimetrics. The width of the loop is held constant at 2.5 mm and the length is varied
47
from 0.25 mm to 10 mm. Figure 17 is a graph of the variation in the inductance of the loop as the
length is increased.
25
D C In d u c t a n c e ,n H
20
15
10
5
0
0
1
2
3
4
5
6
L e n g th , m m
7
8
9
10
Figure 17: Constant Loop Width With Varying Length
When the loop is made longer, the self-inductance increases because there are more
magnetic field lines surrounding the conductor. But there is also more mutual inductance. The
sides of the conductor's loop are in odd mode, that is, their currents are in opposite directions.
The mutual inductance would reduce the loop self-inductance. Figure 17 shows that the
inductance increases as the length increases.
48
Table 8: Comparison of Calculated Mutual and Simulated Inductance
Length
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
mm
0.250
0.738
1.225
1.712
2.200
2.688
3.175
3.663
4.150
4.638
5.125
5.612
6.100
6.588
7.075
7.563
8.050
8.537
9.025
9.512
10.000
1
Q3D
Equ. (82)
Difference
Increment Increment
%
L, nH
Mutual L, nH
1.509
1.146
1.027
0.984
0.970
0.944
0.967
0.948
0.947
0.907
0.938
0.938
0.936
0.933
0.923
0.938
0.906
0.968
0.948
0.866
2
-0.015
0.017
0.045
0.069
0.090
0.107
0.122
0.135
0.146
0.156
0.165
0.174
0.181
0.188
0.194
0.200
0.206
0.211
0.216
0.221
3
100.99
98.52
95.62
92.99
90.72
88.67
87.38
85.76
84.58
82.80
82.41
81.45
80.66
79.85
78.98
78.68
77.26
78.20
77.22
74.48
4
Table 8 shows the reason for the increasing inductance for an increasing length loop. The
width of the loop is held constant while the length is varied from 0.25 mm to 10 mm. Column 1
shows the length of the loop. Column 2 has the increments of the simulated (in Q3D) loop selfinductance. The increment is the size of the step in inductance for each step in length. For
example, row 1 and row 2 are one step. The values of simulated inductance at 0.250 mm and
0.783 mm are subtracted to give 1.509 nH in column 2, row 2. Column 3 has the increments in
mutual inductance. The mutual inductance is calculated using Equation 82. Column 4 shows the
percent difference in the size of the increments in column 2 and column 3.
49
For increasing length, the incremental increase in loop self-inductance is much greater than
the increment of the partial mutual inductance. As the length of the loop is increased, the number
of magnetic loops coupling the sides of the loop will be increased. Since the currents in the sides
is flowing in opposite directions, the mutual inductance will reduce the loop self-inductance. In
this case, the increase in inductance from the increasing length is much greater than the reduction
in inductance due to the mutual inductance.
3.4.3 Constant Loop Length With Varying Width
The loop self-inductance is simulated as the width of the loop is varied using Q3D Extractor
Optimetrics. The length of the loop is held constant at 5 mm and the width is varied from 1 mm
to 5 mm. Figure 18 is a graph of the variation in the inductance of the loop as the width is varied.
2 0 .0 0
In d u c ta n c e , n H
1 8 .0 0
1 6 .0 0
1 4 .0 0
1 2 .0 0
1 0 .0 0
8 .0 0
1 .0 0
1 .5 0
2 .0 0
2 .5 0
3 .0 0
W id t h , m m
Figure 18: Constant Loop Length With Varying Width
3 .5 0
4 .0 0
4 .5 0
5 .0 0
50
3.4.4 Constant Loop Area Held Constant With Varying Length and Width
The inductance was simulated using Q3D Extractor for a varying width and length with the
area of the loop held constant.
Simulation frequency 1 MHz
Loop area 12.5 mm2
Loop length 0.25 to 10mm
Loop width 50 mm to 1.25 mm
Table 9 shows the loop self-inductance while the loop length is increased and the loop width
is decreased to maintain a constant loop area. Figure 19 is a plot of the data. Note that the curve
in Figure 19 shows the inductance decreasing until the length and width are about equal, that is,
the loop is square. Further increases in length increase the loop mutual inductance and thus
reduce the loop self-inductance. At the same time the width is decreasing, further
60
In d u c ta n c e , n H
50
40
30
20
10
0
1
2
3
4
5
6
L e n g th , m m
Figure 19: Inductance for Variable Length and Constant Area
7
8
9
10
51
Table 9: Comparison of Simulated and Calculated Loop Self-Inductance
Length
mm
0.25
0.74
1.23
1.71
2.20
2.69
3.18
3.66
4.15
4.64
5.13
5.61
6.10
6.59
7.08
7.56
8.05
8.54
9.03
9.51
10.00
Width
mm
50.00
16.95
10.20
7.30
5.68
4.65
3.94
3.41
3.01
2.70
2.44
2.23
2.05
1.90
1.77
1.65
1.55
1.46
1.39
1.31
1.25
Simulated
nH
51.67
24.73
17.90
14.93
13.40
12.69
12.35
12.30
12.44
12.68
12.97
13.35
13.79
14.24
14.74
15.15
15.66
16.14
16.70
17.23
17.63
Calculated
nH
48.45
24.37
17.67
14.73
13.26
12.52
12.21
12.16
12.28
12.51
12.83
13.20
13.61
14.05
14.51
14.99
15.47
15.96
16.46
16.96
17.46
%
Difference
-6.23
-1.42
-1.26
-1.34
-1.11
-1.30
-1.14
-1.14
-1.25
-1.33
-1.07
-1.11
-1.30
-1.31
-1.54
-1.10
-1.23
-1.09
-1.45
-1.54
-0.94
3.4.5 Analytical Formula
The resistance and loop self-inductance for a single loop of constant dimensions are shown
in Table 7. The simulated and calculated loop self-inductance differ by about 1.4% [13].
52
M
2
2

0 l   l
d
  1   l    1  d  
ln

2   d
d  
l l

 
M
=
Mutual inductance, H
l
=
Length of conductor, m
d
=
Distance between conductors, m




3.5 Overview of Chapter 3
In chapter 3 simple electromagnetic structures were presented and analyzed. The results
were compared to calculated values from analytical formulas.
(82)
53
Chapter 4
APPLICATIONS
4 APPLICATIONS
Section 4.1 has the application of loop sensors to traffic control. Section 4.2 shows two
loops being simulated to demonstrate sensing of the presence of a car. Section 4.3 shows an
application to ball grid arrays. In section 4.4 parallel plates are simulated and the result is
compared to published data.
4.1 Application to Inductive Loop Sensors
Inductive loop sensors detect the presence of a car. When the car's presence is detected, the
controller will cause the traffic lights to allow a left turn or to cross a busy street. There are many
types of vehicle sensors, but only the inductive loop is being considered here.
4.1.1 Theory
A loop of conductive wire is buried in a cut that is near the road surface. An inductive loop
is about 1.8 m wide and about 1.8 to 21 m long. Its operating frequency is 10 kHz–200 kHz [13].
When a car is above the sensor loop, eddy currents are induced in the car's conductive metal. The
magnetic field that is created opposes the field of the sensor loop, reducing the loop's inductance.
When a car is above the loop, its ferrous material acts like the iron core in a choke, a
condition called the ferromagnetic effect. The inductance of the loop is increased slightly. This
effect is not as significant as the eddy effect, due to the conductivity of the metal, that is more
significant [13].
The sensor loop is laid in a groove in the pavement. There is some capacitive coupling
between the loop and the pavement. The amount of coupling depends upon the dielectric
54
constant of the sealant used to cover the loop. This parasitic capacitance appears in parallel with
the loop inductance, thus reducing the impedance of the loop circuit.
A sensor loop can be tested using a short-circuited circle of wire, about 10 cm in diameter.
Ideally, the test loop would approximate the size and shape of the underside of a car.
A horizontal conducting plate, with the same dimensions as a car's underbody, can be used
to model a car. A loop of conducting wire, with the dimensions of a car's underbody can also be
used to model a car.
Loop sensitivity, SL, expresses the amount of change in inductance when a car is above the
sensor loop [13].
S L  100 
L NV  LV
L NV
LNV
= Inductance with no vehicle above loop, H
LV
= Inductance with vehicle above loop, H
SL 
AV 1 2 F 1
A d 212 F 2
AV
=
Area of car's underside, m2
d21
=
Distance of test loop above sensor loop, m
1
2
 Sensor loop length, m
 Length of shorted loop, m
F1 & F2 = Factors that correct for nonuniform loop flux
Sensitivity is reduced if the car's area is smaller than the sensor loop's area. Sensitivity is
inversely proportional to the square of the distance from the car to the sensor loop.
External inductance of a rectangular loop is [13]:
(83)
(84)
55
Le0  Lep1  Lep 2
Le0
= External loop inductance, H
Lep1
= Inductance of crosswise conductors, H
Lep2
= Inductance of lengthwise conductors, H
(85)
Self-inductance of a rectangular loop is given by:
L 0  L 0  L0
i
e
L0
= Self-inductance of rectangular loop, H
Li0
= Internal inductance of rectangular loop, H
Le0
= External inductance of rectangular loop, H
Li0  21  2  Li
(86)
(87)
Li = Internal inductance, H
l1 and l2
= Width and Length of the loop, m
4.2 Simulate Two Loops
The loop was simulated using Q3D Extractor. Figure 20 shows the model, resized to show
the design.
56
Figure 20: Two Loops.
A simulation was tried with a solid copper plate above the sensor loop. Q3D Extractor's
result matrix shows only the loop inductance of the lower loop. The loop mutual inductance will
be added to the loop inductance of the lower loop, as if the current in the plate were flowing in
the same direction as the current in the lower loop. The simulator will only show the self- and
mutual inductance separately for a net that has a source and sink. Any arrangement of source and
sink on a conducting plate would give incorrect results.
Assuming that since copper is a good conductor the current in the upper plate is forced to
the outside by the skin effect. Thus, using a loop in place of a metal plate gives a reasonably
correct result. A source and sink can be applied to the upper loop when the copper plate is
replaced by a conducting loop. The result matrix will then show the loop self- and mutual
inductances separately, allowing the correct loop inductance to be calculated.
Referring to Table 10, the calculations are for round, #14 wire. The square wire has the
same area as #14 wire. The loops are so large (1.8 m2) that using square wire will introduce little
error. The loop size is 6 ft. x 6 ft. and the upper loop is at 4 in. height. Four inches is a reasonable
height for a car's underbody.
57
Table 10: Loop Dimensions
Wire gauge
# 14
Conductor size
Loop size
1.443 mm x 1.443 mm 1.83 m x 1.83 m
Loop separation
101.6 mm
The equations in [13] do not take frequency into account, The text says elsewhere that the
typical frequency range for loop sensors is 10 kHz–200 kHz. Low frequency simulations are
done in Q3D Extractor and the "DC" results are used.
Table 11: Single Loop Results
Single Loop
DC Resistance
DC Loop Inductance
68.44 m
10.49 H
Table 11 shows the DC resistance and inductance for a single loop. Figure 21 shows a
frequency sweep for the same loop. The sweep is from 1 kHz–26 kHz. This shows the increasing
AC resistance and the decreasing inductance from the skin effect.
Figure 22 is a graph of the simulation of the Q of the loop.
58
A C R e s is t a n c e , m 
120
100
80
60
0
5
10
15
20
25
15
20
25
A C I n d u c t a n c e , H
L e n g th , m m
1 0 .4
1 0 .2
10
0
5
10
L e n g th , m m
Figure 21: AC Resistance and Inductance for a Single Loop.
Q3D
0 .8
0 .7
0 .6
Q
0 .5
0 .4
0 .3
0 .2
0 .1
0
5
Figure 22: Q of a Single Loop.
10
15
F re q u e n c y, k H z
Q3D
20
25
59
Table 12: Two Loop Result
Two Loop
Inductance
DC Resistance
0.06844 
DC Loop Self-
Loop Mutual
Loop
10.526 H
2.239 H
8.286 H
Figure 23 shows the AC resistance and inductance a a function of frequency.
A C I n d u c t a n c e , H
A C R e s is t a n c e , m 
90
80
70
60
0
5
10
15
F re q u e n c y, k H z
20
25
1 0 .5
1 0 .4 5
1 0 .4
0
5
10
15
F re q u e n cy, k H z
20
25
Figure 23: AC Resistance and Inductance for Two Loops.
Figure 24 is the self and mutual inductance for the same two loops. Both loops are signal
loops, allowing Q3D Extractor to simulate the self- and mutual inductances of the loops.
60
I n d u c t a n c e , H
- 3 .1 8 7 5
- 3 .1 8 8 0
- 3 .1 8 8 5
- 3 .1 8 9 0
0
5
10
15
F re q u e n cy, k H z
20
25
20
25
I n d u c t a n c e , H
1 0 .4 2
1 0 .4 0
1 0 .3 8
0
5
10
15
F re q u e n c y, k H z
Figure 24: Mutual Inductance(Dashed Line) and Self Inductance(Solid Line) for Two Loops.
Figure 25 shows the Q of two active loops as a function of frequency.
0 .8
0 .7
0 .6
Q
0 .5
0 .4
0 .3
0 .2
0 .1
0
5
Figure 25: Q of Two Loops. Q3D
10
15
F re q u e n c y, k H z
20
25
61
The loop inductance with no car present is 10.526 H and the loop inductance with a car
present is 8.286 H.
Table 13: Single Loop With Metal Plate
Single Loop, Metal Plate
DC Resistance
DC Loop Inductance
60.064 m
10.488 H
Table 13 is the DC resistance and inductance of a single loop and a conducting plate that is
101.6 mm above it. The plate 1 mm thick and is of copper. Figure 26 is a frequency sweep of the
A C I n d u c t a n c e , H
A C R e s is t a n c e , m o h m
low frequency AC resistance and inductance. The sweep is from 1 kHz–26 kHz.
100
80
60
0
5
10
15
F re q u e n c y, k H z
20
25
10
15
F re q u e n cy, kH z
20
25
9 .0
8 .8
8 .6
8 .4
0
5
Figure 26: Low Frequency Resistance (Solid) and Inductance (Dashed) of Single Loop With
a Conducting Plate. Simulated Using Q3D Extractor.
62
0 .5 0
0 .4 0
Q
0 .3 0
0 .2 0
0 .1 0
0
5
10
15
F re q u e n cy, kH z
20
25
Figure 27: Q for Single Loop With a Conducting Plate.
Figure 27 shows the Q for a frequency sweep of the single loop and conductive plate.
Table 14: DC Results for Two Loops With One Shorted
Two Loops, One Shorted
DC Resistance DC Loop Inductance
60.064 m
10.488 H
Figure 28 shows the AC resistance and inductance for two loops with the upper loop short
circuited. Figure 29 is a graph of the simulation of the Q for the same two loops as they are swept
for frequency.
A C I n d u c t a n c e , H
A C R e s is ta n c e , m o h m
63
90
80
70
60
0
5
10
15
F re q u e n c y, k H z
20
25
10
15
F re q u e n cy, kH z
20
25
9 .7
9 .6
9 .5
9 .4
9 .3
0
5
Figure 28: AC Resistance and Inductance for Two Loops With One Shorted. Q3D
0 .4 0
Q
0 .3 0
0 .2 0
0 .1 0
0 .0 0
0
5
10
15
F re q u e n cy, kH z
Figure 29: Q for Two Loops With one Shorted. Q3D
20
25
64
Figures 30 and 31 show field overlays from HFSS
.
Figure 30 shows the surface current density (J, A/m) for the sensor loop and the upper
metal plate. Notice in Figure 30 the current in the plate is circulating counterclockwise and the
current in the sensor loop is circulating clockwise.
Figure 30: Current Density.
Figure 31 shows the 2D Extractor simulation of the electric field around the sensor loop.
When this field penetrates the plate, an eddy current is generated. An H-field in the opposite
direction opposes the original field, reducing the loop inductance of the sensor loop.
65
Figure 31: Electric Field.
4.3 Application to Ball Grid Arrays
It is often believed that clearance holes in printed circuit boards will substantially increase
the inductance of the power plane. Many holes occur when, for example, a ball grid array is used.
The effect on a parallel plate transmission line of many holes is demonstrated and compared
to a similar transmission line which has no holes. The parallel plate transmission line is shown in
Figure 32. This is the transmission line that has no holes, other than some vias.
The transmission line consists of two copper plates separated by an air dielectric. The front
of the parallel plate has two vias–the upper is the source and the lower is the sink. These vias are
shown in Figure 33. The input and output vias are embedded in the conducting planes and are
separated by the dielectric. The upper and lower conductors are connected together at the back of
the structure by a single blind via that is entirely within the dielectric as shown in Figure 34.
66
Figure 32: Parallel Plate Transmission Line.
Figure 33: Front, Showing Input and Output Vias.
Figure 34: Back, Showing Blind Via Connecting Conductors.
67
Figure 35 and a detailed view of the lower right hand corner, Figure 36, shows a 3D
Extractor simulation of a parallel plate with a field of holes. The field overlay is of the surface
current ( JAC A/m). The The simulation was done at 1.75 GHz.
Figure 35: Field Overlay Showing Surface Current Density, J A/m
68
Figure 36: Detail Field Overlay Showing Surface Current Density
4.4 Simulate Parallel Plate With Vias Only and Parallel Plate With Field of Holes
The inductances of two parallel plate current loops are compared. The first one has two
solid plates, as seen in Figure 37. The other is perforated by a field of holes, as seen in figure 38.
Both have a source via in the front of the top plate and a sink via in the lower plate. The plates
are connected at the end with a blind via at the opposite end.
69
Figure 37: Solid Parallel Plate Current Loop.
Figure 38: Parallel Plate With a Field of Holes.
Both the solid and perforated plates are 250 mils square with vias of 5 mils radius. The
conductors are 3 mils thick and the air dielectric is 2 mils thick. The holes in the perforated
conductor have a radius of 10 mils and are spaced in 25 mil increments, with a total of 68 holes.
The DC simulated inductance and resistance are shown in Table 15. The simulated results
are compared with previously published data [2].
70
Table 15: DC Resistance and DC Inductance for a Plane and Field of Holes
Plane
Simulated
Holes
DC R
DC L
DC R
DC L
0.9593 mohm
195.45 pH
1.530 mohm
279.69 pH
A C R e s is t a n c e , m o h m
Bogatin
192 pH
243 pH
0 .9 8 0
0 .9 7 8
0 .9 7 6
0 .9 7 4
0 .9 7 2
500
550
600
650
F re q u e n cy, kH z
700
750
550
600
650
F re q u e n cy, kH z
700
750
In d u c ta n c e , p H
1 9 4 .7
1 9 4 .6
1 9 4 .5
1 9 4 .4
500
Figure 39: Sweep of AC R and L for Solid Parallel Plate. Simulated Using Q3D Extractor
In d u c ta n c e , p H
A C R e s is t a n c e , m o h m
71
1 .5 2 0
1 .5 1 0
1 .5 0 0
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
F re q u e n c y, k H z
0 .7
0 .8
0 .9
1
0 .2
0 .3
0 .4
0 .5
0 .6
F re q u e n c y, k H z
0 .7
0 .8
0 .9
1
2 7 7 .0
2 7 6 .0
2 7 5 .0
0 .1
Figure 40: Sweep of AC R and L for Field of Holes. Simulated Using Q3D Extractor
4.4.1 Analysis
The calculated inductance is compared to the published value in Bogatin [2]. The values for
the solid plate are within 1.8%. The values for the field of holes differs by 13.1%.
4.5 Overview of Chapter 4
In chapter 4 the application of various simple electromagnetic structures were simulated.
The application of these structures were discussed.
72
Chapter 5
CONCLUSIONS
5 CONCLUSIONS
Electromagnetic principles along with analytical formulas have been presented. The skin
effect, impedance of electromagnetic materials as a function of frequency along with formulas
for calculation of these characteristics has been presented.
Electromagnetic structures related to these concepts have been simulated and analyzed using
full-wave electromagnetic simulation software. Analytic formulas for these structures are used to
compare calculated with simulated values. Practical applications utilizing these simulated
structres were presented and compared to published results.
73
APPENDIX
% TWO_PORT_TEE.m
% READS .S2P FILE FROM HFSS SIMULATION. MAKES ABCD PARAMs
% CALCS 2-PORT T-CKT. CONVERTS RESULT BACK TO abcd PARAMs
% THEN TO S-PARMs FOR PLOTTING (OF S11 & S12).
clear all;
Z0=50;
% Zs & Zl don't matter
Zs=2.9714;
Zl=2.9714;
fprintf('\n\n****************\nThis version is for T type 2-port
ckt\n******************\n\n')
fprintf('\n\nZs= %s, Zl= %s, Z0=
%s\n\n',num2str(Zs),num2str(Zl),num2str(Z0))
fprintf('.s2p MUST be Z0=50 ohm, ref Z will be converted ')
fprintf('to %s, ohms for calculations\n\n',num2str(Z0))
% change this to name of .s2p file
AA = read(rfdata.data,'ABC.s2p');
% .s2p file must have Z0=50 ohm. If not, 50ohm is
% presumed
% convert to new Z0
analyze(AA,AA.Freq,Zl,Zs,Z0)
% MAKE Z-MATRIX
ZZ=s2z(AA.S_Parameters,Z0);
% make frequency matrix
freak(1,:) = AA.Freq;
% make ABCD params @ Z0
ABCD = s2abcd(AA.S_Parameters,Z0);
% pick out ABCD params
for nn=1:size(freak,2)
A(1,nn)=ABCD(1,1,nn);
B(1,nn)=ABCD(1,2,nn);
C(1,nn)=ABCD(2,1,nn);
D(1,nn)=ABCD(2,2,nn);
end
% Z1,Z2,Z3 ARE COMPONENTS OF POZAR's T-CKT
for nn=1:size(freak,2)
Z3(1,nn)=(1/C(1,nn));
Z1(1,nn)=((A(1,nn)-1)*Z3(1,nn));
Z2(1,nn)=((D(1,nn)-1)*Z3(1,nn));
74
end
% MAKE 2-PORT CKT INTO ABCD PARAMS
for nn=1:size(freak,2)
abcd(1,1,nn)=1+(Z1(1,nn)/Z3(1,nn));
abcd(1,2,nn)=Z1(1,nn)+Z2(1,nn)+((Z1(1,nn)*Z2(1,nn))/Z3(1,nn));
abcd(2,1,nn)=1/Z3(1,nn);
abcd(2,2,nn)=1+(Z2(1,nn)/Z3(1,nn));
end
% make s matrix
s=abcd2s(abcd,Z0);
PP=rfdata.data('S_Parameters',s,'Freq',freak,'Z0',Z0, 'ZS',Z0, 'ZL',Z0)
write(PP,'hold_s.s2p','MA')
z0=Z0
DUT=rfckt.datafile;
read(DUT,'hold_s.s2p');
analyze(DUT,freak,z0,z0,z0);
in_line=rfckt.microstrip('EpsilonR',1.0001, 'Height',.050e-3,...
'LineLength',.635e-3,'LossTangent',0,'Thickness',.0508e3,'Width',.00635,'SigmaCond',inf);
analyze(in_line,freak,z0,z0,z0);
out_line=rfckt.microstrip('EpsilonR',1.0001, 'Height',.050e-3,...
'LineLength',.635e-3,'LossTangent',0,'Thickness',.0508e3,'Width',.00635,'SigmaCond',inf);
analyze(out_line,freak,z0,z0,z0);
all=rfckt.cascade('Ckts',{in_line,DUT,out_line});
analyze(all,freak,z0,z0,z0);
info=calculate(all,'S11','S12','none');
[s11,s12]=deal(info{1},info{2});
figure(1)
clf
plot(freak,abs(s11))
grid
title('s11')
figure(3)
clf
plot(freak,abs(s12),'r')
grid
title('s12');
for nn=1:size(freak,2)
omegaa(1,:)=freak(1,:)*2*pi;
% calc L&C for each element
75
% L=Xl/omega
L(1,nn)=imag( Z1(1,nn) /omegaa(1,nn) );
L(2,nn)=imag( Z2(1,nn) /omegaa(1,nn) );
L(3,nn)=imag( Z3(1,nn) /omegaa(1,nn) );
% C=1/(w*Xc)
Cap(1,nn)= imag( 1/ ( Z1(1,nn) *omegaa(1,nn) ) );
Cap(2,nn)= imag( 1/ ( Z2(1,nn) *omegaa(1,nn) ) );
Cap(3,nn)= imag( 1/ ( Z3(1,nn) *omegaa(1,nn) ) );
end
clear nn ans ZinPAR Zl
76
% 2_PORT_PI.m
% REFERENCE: POZAR, pg.185,
% Convert ABCD to 3-element PI ckt.
% ref. POZAR, pg 185.
clear all;
fprintf('\nDOES Zs & Zl HAVE TO MATCH DESIRED Z0 ???')
% These may matter for analyze, Z0 certainly does.
Zs= 50;
% don't care
Zl= 50;
% don't care Yparams; don't care Y1,2,3
Z0= 50; % match HFSS; don't care Y1,2,3
fprintf('\n\n****************\nThis version is for PI type 2-port
ckt\n******************\n\n')
fprintf('\n\nZs= %s, Zl= %s, Z0=
%s\n\n',num2str(Zs),num2str(Zl),num2str(Z0))
fprintf('.s2p MUST be Z0=50 ohm, ref Z will be converted ')
fprintf('to %s, ohms for calculations\n\n',num2str(Z0))
AA = read(rfdata.data,'PI_RL.s2p');
% (Z0 must be specified in .s2p
% Z0 must!! be 50 ohm, else S-params will be for
% whatever Z0 but MATLAB assumes it is for 50 ohm)
% convert to new Z0
analyze(AA,AA.Freq,Zl,Zs,Z0)
% MAKE FREQUENCY-MATRIX
freak(1,:) = AA.Freq;
% make ABCD params @ Z0
ABCD = s2abcd(AA.S_Parameters,Z0);
% pick out ABCD params
for nn=1:size(freak,2)
A(1,nn)=ABCD(1,1,nn);
B(1,nn)=ABCD(1,2,nn);
C(1,nn)=ABCD(2,1,nn);
D(1,nn)=ABCD(2,2,nn);
end
% MAKE Y PARAMS
for nn=1:size(freak,2)
Y(3,nn)= 1/B(1,nn);
Y(2,nn)=(A(1,nn)-1)*Y(3,nn);
Y(1,nn)=(D(1,nn)-1)*Y(3,nn);
end
77
% MAKE Y MATRIX
for nn=1:size(freak,2)
Yparms(:,:,nn)=abcd2y(ABCD(:,:,nn));
end
% CALCULATE CAPACITANCE & INDUCTANCE
for nn=1:size(freak,2)
omegaa=2*pi*freak(1,nn);
% C=Yc/(i*w)
Cap(1,nn)= imag(Y(1,nn))/(omegaa);
Cap(2,nn)= imag(Y(2,nn))/(omegaa);
Cap(3,nn)= imag(Y(3,nn))/(omegaa);
% L=1/(i*w*Yl)
L(1,nn)=imag(1/((Y(1,nn))*omegaa));
L(2,nn)=imag(1/((Y(2,nn))*omegaa));
L(3,nn)=imag(1/((Y(3,nn))*omegaa));
end
fprintf('\nOnly the positive values for cap & L are valid\n')
fprintf('Each column (set of values) must have consistent signs for \n')
fprintf('cap & inductance\n')
78
REFERENCES
[1] Bansevicius, R., Virbalis, J. A., "Distribution of Electric field in the Round Hole of Plane
Capacitor", Journal of Electrostatics, 64 (2006) 226-233, Elsevier B.V.
[2] Bogatin, E., Signal Integrity Simplified Upper Saddle River, New Jersey: Prentice Hall
Professional Technical Reference, 2004
[3] Chapra, S. C., Canale, R. P., Numerical Methods for Engineers, Fifth Ed., McGraw Hill New
Your, 2006
[4] Edminister, J. A., Schaum's Theory and Problems, Electromagnetics, Second Edition,
McGraw-Hill, New York, 1993.
[5] Koshkin, N..I., Shirkevich, M.G, Handbook of Elementary Physics, Mir Publishers, Moscow,
1982.
[6] Johnson, H.W., Graham, M., High-Speed Digital Design, A Handbook of Black Magic,
Prentice Hall PTR, Upper Saddle River, New Jersey, 1993.
[7] Landee, R.W., Davis, D.C., Albrecht, A.P., Electronic Designers' Handbook, McGraw-Hill
Book Company, Inc., New York, 1957.
[8] Laney, O., "Analysis and Application of Transmission Line Conductors," M.S. thesis, Dept.
Elect. Eng., San Jose State Univ., San Jose, CA, 2013.
[9] Miner, G.F., Lines and Electromagnetic Fields for Engineers, Oxford University Press, Inc.,
New York, 1996.
[10] Pozar, D.M., Microwave Engineering, Third Edition, John Wiley and Sons, New York,
2005
[11] Ramo, S., Whinnery, J. R., Fields and Waves in Modern Radio, John Wiley and Sons, New
York, 1959
79
[12] Sadiku, M. N. O., Numerical Techniques in Electromagnetics, Second Ed., CRC Press, Boca
Raton, FL, 2001
[13] Traffic Detector Handbook: Third Edition–Volume I, Publication No. FHWA-HRT-06-108,
Research, Development, and Technology, Turner-Fairbank Highway Research Center, McLean,
VA., 2006.
[14] Ulaby, F.T., Fundamentals of Applied Electromagnetics, 2001 Media Edition, Prentice-Hall,
Upper Saddle River, New Jersey, 2001.
[15] Wadell, B,C., Transmission Line Design Handbook, Artech House, Inc., Norwood,
Massachusetts, 1991.
[16] Wheeler, H. A., "Formulas for the Skin Effect", Proceedings of the I.R.E., vol. 30, no. 9, pp.
412-424,1942