Samenvatting
4-Telecommunication-1
Inhoudopgaven
Math consepts.......................................................................................................................................4
Cross product...................................................................................................................................4
Dot product......................................................................................................................................4
Differential (afgeleide).....................................................................................................................4
Del (nabla).......................................................................................................................................4
Electric concepts...................................................................................................................................4
Permittivity......................................................................................................................................4
Permeability.....................................................................................................................................5
Names..............................................................................................................................................5
Maxwell’s equations........................................................................................................................5
Electrostatics & magneto statics...........................................................................................................5
Electrostatics (part1: nabla cdot D = %rho_v).....................................................................................6
Charge and current distribution.......................................................................................................6
Charge density.................................................................................................................................6
Current density.................................................................................................................................7
Coulomb’s law.................................................................................................................................7
Gauss’s law......................................................................................................................................8
The differential (afgeleide) form of Gauss’s Law:......................................................................8
The integral form of Gauss’s law:...............................................................................................8
Electrostatics (part2: nabla times E = 0)...............................................................................................9
Formula............................................................................................................................................9
Electric Scalar Potential...................................................................................................................9
Electric potential............................................................................................................................10
Conductors and Dielectrics............................................................................................................11
Dielectrics..................................................................................................................................11
Magnetostatics....................................................................................................................................12
In a current carrying conductor......................................................................................................13
The Biot-Savart Law......................................................................................................................13
Infinity long wire......................................................................................................................13
Forces between 2 parallel conductors............................................................................................13
Gauss’s Law (D vs B) (surface).....................................................................................................14
Ampere’s Law (E & H) (contour)..................................................................................................14
Magnetic field intensity in a wire..................................................................................................14
Magnetic field intensity in a toroidal coil (torus, toroid)...............................................................15
Vector magnetic Potential A..........................................................................................................15
Magnetic boundary conditions.......................................................................................................15
Boundary Conditions..........................................................................................................................16
Electro............................................................................................................................................16
Examples...................................................................................................................................16
Magnetics.......................................................................................................................................17
Examples...................................................................................................................................17
,Resume...............................................................................................................................................18
Examples Electro/ magneto statics.....................................................................................................19
Example 1......................................................................................................................................20
Example 2......................................................................................................................................23
Example 3......................................................................................................................................25
Example 4......................................................................................................................................26
Time varying.......................................................................................................................................27
Dynamic fields...............................................................................................................................27
Faraday’s Law................................................................................................................................28
Equation....................................................................................................................................28
Example Faraday’s Law............................................................................................................29
Gauss’s Law...................................................................................................................................30
Equations...................................................................................................................................30
Ampère’s Law................................................................................................................................30
Equations...................................................................................................................................30
Plane Wave Propagation.....................................................................................................................31
Unbounded EM Waves..................................................................................................................31
Wave equasion...............................................................................................................................31
Lossless media...............................................................................................................................32
Propagation...............................................................................................................................32
Lossy media...................................................................................................................................33
Propagation...............................................................................................................................33
Lossy and lossless media simplified..............................................................................................34
Power Density................................................................................................................................34
Resume formules...........................................................................................................................35
Wave reflection and transmission.......................................................................................................36
Types of reflections........................................................................................................................36
Boundary behavior.........................................................................................................................36
In a lossless media.....................................................................................................................37
Coefficients..........................................................................................................................37
Power flow...........................................................................................................................38
In a lossy media.........................................................................................................................39
Coefficients..........................................................................................................................39
Snell’s Law....................................................................................................................................40
Snell’s Law of reflection...........................................................................................................40
Snell’s Law of refraction...........................................................................................................40
Index of refraction.....................................................................................................................41
Reflectivity and Transmissivity.....................................................................................................42
Power density S and power P....................................................................................................42
Reflection coefficient GAMMA in an orthogonal (90°) system...............................................42
Reflection coefficient GAMMA in a parallel (0°) system........................................................42
Transmissivity T (90° & 0°)......................................................................................................42
Difference ortho and parallel....................................................................................................43
Example.........................................................................................................................................43
Transmission Lines.............................................................................................................................45
Wavelength....................................................................................................................................46
Transmission line effects...........................................................................................................46
Transverse electromagnetic (TEM) transmission lines..................................................................47
Lumped-Element Model................................................................................................................47
Telegrapher’s equasion..............................................................................................................49
4-Telecommunication-1 2
, Telegrapher’s equasion phasor form.........................................................................................49
4-Telecommunication-1 3
, Math consepts
Cross product
a×b=‖a‖∗‖b‖∗sin( θ )=(
| || || |
ya z0 za x0 xa
, ,
yb zb zb xb xb
y0
yb
)
If a×b=1 then we have a new vector at 90° of a and b.
Dot product
a⋅b=‖a‖∗‖b‖∗cos( θ )=a x b x +a y b y + a z b z
If a⋅b=0 then we have 2 vectors a and b that are perpendicular.
Differential (afgeleide)
d
dx
Del (nabla)
⃗ is a multipurpose differential operator
Del ∇
⃗
∇=i ∂ + j ∂ +k ∂ Definition
∂x ∂ y ∂z
⃗ A=i ∂ A + j ∂ A + k ∂ A
∇
The Gradient (De stijlheid van iets
∂x ∂y ∂z voorgesteld door een vector)
⃗ ∂ Ax ∂ A y ∂ Az The Divergence (Het is de neiging van
∇⋅A=i +j +k een veld om naar een punt toe te
∂x ∂y ∂z
convergeren of ervan af te wijken)
⃗ A=i( ∂ A y + ∂ A x )+ j( ∂ A z + ∂ A y )+ k ( ∂ A x + ∂ A z )
∇×
The Curl (Het geeft een vector dat
∂x ∂ y ∂ y ∂z ∂z ∂x weergeeft hoe de neiging is van het veld
om rond een punt te circuleren)
Electric concepts
Permittivity
Permittivity ε is the propagation of an electric field in a material. The formula is defined in vacuum.
ε= εr ε0
εr = This is the resistance of the used material
ε0 = This is the resistance in vacuum =8.854∗10−12 Fm−1
4-Telecommunication-1 4
4-Telecommunication-1
Inhoudopgaven
Math consepts.......................................................................................................................................4
Cross product...................................................................................................................................4
Dot product......................................................................................................................................4
Differential (afgeleide).....................................................................................................................4
Del (nabla).......................................................................................................................................4
Electric concepts...................................................................................................................................4
Permittivity......................................................................................................................................4
Permeability.....................................................................................................................................5
Names..............................................................................................................................................5
Maxwell’s equations........................................................................................................................5
Electrostatics & magneto statics...........................................................................................................5
Electrostatics (part1: nabla cdot D = %rho_v).....................................................................................6
Charge and current distribution.......................................................................................................6
Charge density.................................................................................................................................6
Current density.................................................................................................................................7
Coulomb’s law.................................................................................................................................7
Gauss’s law......................................................................................................................................8
The differential (afgeleide) form of Gauss’s Law:......................................................................8
The integral form of Gauss’s law:...............................................................................................8
Electrostatics (part2: nabla times E = 0)...............................................................................................9
Formula............................................................................................................................................9
Electric Scalar Potential...................................................................................................................9
Electric potential............................................................................................................................10
Conductors and Dielectrics............................................................................................................11
Dielectrics..................................................................................................................................11
Magnetostatics....................................................................................................................................12
In a current carrying conductor......................................................................................................13
The Biot-Savart Law......................................................................................................................13
Infinity long wire......................................................................................................................13
Forces between 2 parallel conductors............................................................................................13
Gauss’s Law (D vs B) (surface).....................................................................................................14
Ampere’s Law (E & H) (contour)..................................................................................................14
Magnetic field intensity in a wire..................................................................................................14
Magnetic field intensity in a toroidal coil (torus, toroid)...............................................................15
Vector magnetic Potential A..........................................................................................................15
Magnetic boundary conditions.......................................................................................................15
Boundary Conditions..........................................................................................................................16
Electro............................................................................................................................................16
Examples...................................................................................................................................16
Magnetics.......................................................................................................................................17
Examples...................................................................................................................................17
,Resume...............................................................................................................................................18
Examples Electro/ magneto statics.....................................................................................................19
Example 1......................................................................................................................................20
Example 2......................................................................................................................................23
Example 3......................................................................................................................................25
Example 4......................................................................................................................................26
Time varying.......................................................................................................................................27
Dynamic fields...............................................................................................................................27
Faraday’s Law................................................................................................................................28
Equation....................................................................................................................................28
Example Faraday’s Law............................................................................................................29
Gauss’s Law...................................................................................................................................30
Equations...................................................................................................................................30
Ampère’s Law................................................................................................................................30
Equations...................................................................................................................................30
Plane Wave Propagation.....................................................................................................................31
Unbounded EM Waves..................................................................................................................31
Wave equasion...............................................................................................................................31
Lossless media...............................................................................................................................32
Propagation...............................................................................................................................32
Lossy media...................................................................................................................................33
Propagation...............................................................................................................................33
Lossy and lossless media simplified..............................................................................................34
Power Density................................................................................................................................34
Resume formules...........................................................................................................................35
Wave reflection and transmission.......................................................................................................36
Types of reflections........................................................................................................................36
Boundary behavior.........................................................................................................................36
In a lossless media.....................................................................................................................37
Coefficients..........................................................................................................................37
Power flow...........................................................................................................................38
In a lossy media.........................................................................................................................39
Coefficients..........................................................................................................................39
Snell’s Law....................................................................................................................................40
Snell’s Law of reflection...........................................................................................................40
Snell’s Law of refraction...........................................................................................................40
Index of refraction.....................................................................................................................41
Reflectivity and Transmissivity.....................................................................................................42
Power density S and power P....................................................................................................42
Reflection coefficient GAMMA in an orthogonal (90°) system...............................................42
Reflection coefficient GAMMA in a parallel (0°) system........................................................42
Transmissivity T (90° & 0°)......................................................................................................42
Difference ortho and parallel....................................................................................................43
Example.........................................................................................................................................43
Transmission Lines.............................................................................................................................45
Wavelength....................................................................................................................................46
Transmission line effects...........................................................................................................46
Transverse electromagnetic (TEM) transmission lines..................................................................47
Lumped-Element Model................................................................................................................47
Telegrapher’s equasion..............................................................................................................49
4-Telecommunication-1 2
, Telegrapher’s equasion phasor form.........................................................................................49
4-Telecommunication-1 3
, Math consepts
Cross product
a×b=‖a‖∗‖b‖∗sin( θ )=(
| || || |
ya z0 za x0 xa
, ,
yb zb zb xb xb
y0
yb
)
If a×b=1 then we have a new vector at 90° of a and b.
Dot product
a⋅b=‖a‖∗‖b‖∗cos( θ )=a x b x +a y b y + a z b z
If a⋅b=0 then we have 2 vectors a and b that are perpendicular.
Differential (afgeleide)
d
dx
Del (nabla)
⃗ is a multipurpose differential operator
Del ∇
⃗
∇=i ∂ + j ∂ +k ∂ Definition
∂x ∂ y ∂z
⃗ A=i ∂ A + j ∂ A + k ∂ A
∇
The Gradient (De stijlheid van iets
∂x ∂y ∂z voorgesteld door een vector)
⃗ ∂ Ax ∂ A y ∂ Az The Divergence (Het is de neiging van
∇⋅A=i +j +k een veld om naar een punt toe te
∂x ∂y ∂z
convergeren of ervan af te wijken)
⃗ A=i( ∂ A y + ∂ A x )+ j( ∂ A z + ∂ A y )+ k ( ∂ A x + ∂ A z )
∇×
The Curl (Het geeft een vector dat
∂x ∂ y ∂ y ∂z ∂z ∂x weergeeft hoe de neiging is van het veld
om rond een punt te circuleren)
Electric concepts
Permittivity
Permittivity ε is the propagation of an electric field in a material. The formula is defined in vacuum.
ε= εr ε0
εr = This is the resistance of the used material
ε0 = This is the resistance in vacuum =8.854∗10−12 Fm−1
4-Telecommunication-1 4
