EMTL

Prerequisites

 ElectroMagnetic Waves and Transmission Lines

Fundamentals (prerequisites)

Unit-1: Electrostatics, Magnetostatics

Unit-2: Time Varying Fields

Unit-3,4 : EM wave characteristics

 Unit-5,6: Transmission Lines

Textbook-Author - Mathew N Sadiku

TextbookPDF   +  SolutionsManualPDF

         (both are external links)

Coordinate systems

(click on image to maximize it and press ESC when done)

Legend for the below systems:

Ex: in Rectangular(cartesian) coordinate system (x, y, z) ----> (dx, dy, dz)

 (x, y, z)  are coordinates for the system----> (dx, dy, dz) are their differential elements respectively. 

Rectangular(cartesian) coordinate system (x, y, z) ----> (dx, dy, dz)

Cylindrical coordinate system (ρ, φ, z)  ----> (dρ, ρ dφ, dz)

Spherical coordinate system (r, θ, φ)  ----> (dr, r dθ, r sinθ dφ)

Vector calculus

D = Dx ax + Dy ay + Dz az

dot product (is a scalar)

∇.D = (∂/∂x) Dx  +(∂/∂y) Dy  + (∂/∂z) Dz 

cross product (is a vector)

            | ax          ay          az    |

∇xD = | Dx         Dy         Dz    |

            |(∂/∂x)  (∂/∂y)    (∂/∂z)  |

del operator (is a vector)

∇V =  (∂/∂x) V  ax +  (∂/∂y) V ay  +  (∂/∂z) V az

others

Divergence Theorem:  ∫D ds = ∫ ∇.D dv           ----( surface integral to divergence)

Stoke's theorem: ∫D dl = ∫ ∇xD ds           ----( line integral to curl)

laplacian operator : ∇²

Electrostatics

Charge stationary (So only static electric field around it)

Maxwell's Electro-Static equations are:
∇·D = ρ_v
∇ × E = 0

Magnetostatics

Charge move with constant velocity (a moving charge of constant velocity produces constant current and hence static magnetic field)

Maxwell's Magneto-Static equations are:
∇ × H = J
∇·B = 0 (1d)

Time Varying Fields

When a charge moves with varying velocity, it is associated with varying E and H.
Maxwell's Equations in time-varying fields are:
∇·D = ρ_v
∇ × E = −∂B/∂t
∇ × H = J + ∂D/∂t
∇·B = 0

EM Wave Characteristics

Wave characteristics explain the behavior of EM Waves based on the property of the medium.

Wave equations, Uniform plane wave

Waves incident on conducting plane and Dielectric plane.

Boundary conditions

Transmission Lines

Two wire parallel transmission lines are analyzed.

(i) Types

(ii) Parameters

(iii) Transmission line equations

(iv) Primary and Secondary constants

(v) Expressions for Zo, γ, υp and  υg .

(vi) Infinite line concepts

(vii) Losslessness / Low loss characterization

(viii) Distortion – condition for distortionlessness or minimum attenuation

(ix) Loading- Types

Def: A transmission line is a means of transfer of information from one point to another. It is used to connect Source to Load

I- Types of Transmission Lines:

•          Two-wire parallel transmission lines
•    Coaxial lines
•   Twisted pair of lines
•     Planar line 
•     Wire above conducting line
•     Micro strip line
•    Optical fiber cable

Uses of Transmission Lines:

•   Transfer of energy from one circuit to another
•   Can be used as Circuit Elements (like inductor, capacitor)
•  Can be used as impedance matching (Stubs)
•   Coaxial cables are used in lab and to connect TV to Antennas
•   Micro Strips are used in integrated circuits in which metallic strips connecting electronic elements are deposited on dielectric substrates
•   Twisted pairs and Coaxial lines are used in Computer networking such as Internet and Ethernet
•   Parallel lines for Telephony and Power Transmission
•   Planar lines used to connect TX and Antennas
•  Optical fibers are used to transmit information over long and short distances with negligible attenuation. 

II- Parameters:

The transmission line can be represented or approximated to an equivalent circuit as =>

Hence its parameters are Resistance (R units Ω/Km), Inductance (L units H/Km), Conductance (G units S/Km) and Capacitance (C units F/ Km).

Note: R≠ 1/G. Both are different, i.e. R represents resistance of the wire offered per unit distance while G represents Conductance between two conductors (in practical there should be no conduction between two wires {if conduction between two wires exists directly means there is a short circuit})

Ideally R=0, L=0, G=0 and C=0;

 

III- Primary and Secondary Constants:

 

The R, L, G and C are called PRIMARY constants.

There are other constants called Z_O and γ called as SECONDARY constants (Because these are derived from the Primary constants)

IV- Transmission line equations:

Consider a TXn line of length ‘ l  ’ and V_S,  I_S as voltage and current at the SOURCE end while V_L and I_L as Voltage and Current at LOAD end.

The transmission line equations are obtained as follows:

·      Two solutions for above transmission line equations can be obtained.

(i) Exponential Solution

(ii) Hyperbolic Solution

(i) Exponential Solution for eq (6) and (7) respectively are:

(ii) Hyperbolic Solution for eq (6) and (7) respectively are:

Taking I_L common from Above equations (19) and (20)

i.e.

But the impedance of T-equivalent is

Therefore by definition it can be proved :

VII- Losslessness/ Low Loss Characterization:

=> Lossless TXn lines

  

VIII- Distortion- Condition for Distortion less and Minimum attenuation:

IX- Loading –Types of Loading:

Introducing inductance in series with the TXn line is called Loading.

Such lines are called as Loaded Lines.

Effect/ Types:

(i)                 Continuous Loading: Winding a type of iron around the conductor. This increases inductance. This is an expensive process.

(ii)               Patch Loading: This type of loading uses continuously loaded cable separated by sections of unloaded cable. This is an inexpensive method.

(iii)             Lumped Loading: Loading is introduced at uniform intervals. Hysteresis and Eddy current losses are introduced. Design should be optimal. 

Lossy Line:

Smith Chart: