Syllabus
UNIT – I
Electrostatic Fields: Coulomb‘s Law
and Field Intensity -
Electric Fields due to Continuous Charge Distributions – Line Charge, Surface Charge, Volume Charge - Electric Flux Density - Gauss Law – Applications of Gauss Law – Point Charge, Infinite Line Charge, Infinite Sheet Charge, Uniformly Charged Sphere - Electric Potential - Relations Between E and V - Illustrative Problems.
UNIT – II
Magnetostatic Fields: Biot-Savart Law - Ampere‘s Circuital Law – Applications of Ampere’s Circuit Law – Infinite Line Current, Infinite Sheet of Current - Magnetic Flux Density, Maxwell‘s Equations for Static EM Fields – Magnetic Scalar and Vector Potential - Illustrative Problems.
Syllabus
UNIT – III
Maxwell’s Equations (Time Varying Fields): Faraday‘s Law - Transformer and Motional EMFs – Stationary Loop in Time Varying B Field, Moving Loop in Static B Field, Moving Loop in Time Varying Field - Displacement Current
- Maxwell‘s Equations in Different Final Forms, Illustrative Problems.
UNIT – IV
EM Wave Propagation: Waves in General – Wave Propagation in Lossy Dielectrics – Plane Waves in Lossless Dielectrics – Plane Wave in Free Space
– Plane Waves in Good Conductors - Power and the Poynting Vector - Reflection of a Plane wave at Normal Incidence - Reflection of a Plane wave at Oblique – Parallel Polarization, Perpendicular Polarization - Illustrative Problems.
UNIT – V
Transmission Lines: Transmission Line Parameters – Transmission Line Equations – Input Impedance, SWR and Power – The Smith Chart – Applications of Transmission Lines – Transients on transmission Lines – Microstrip Transmission Lines – Illustrative Problems.
Text Books
References
2nd
Web Links
Unit-1
Electrostatic Fields
Contents
Coulomb’s Law
Contd…
Eq-1
Contd…
Contd…
Contd…
EQ-2
Contd…
Contd…
Contd…
Contd…
Contd…
Contd….
Contd…
Contd…
Electric Fields due to Continuous Charge Distributions
Contd…
Contd…
Line Charge
Contd…
Contd…
To evaluate this, it is convenient that we define α1, α2
and α3
Contd…
Surface Charge
Contd…
Contd…
Due to the symmetry of the charge distribution, for every element 1, there is a corresponding element 2 whose contribution along aρ cancels that of element 1
Contd…
In General
Volume Charge
Contd…
Contd…
0, z) due to the
Contd…
Electric Flux Density
Contd…
Gauss Law
Contd…
Which is the first of the four Maxwell's equations to be derived.
Applications of Gauss Law
Contd…
Point Charge
Contd…
Infinite Line Charge
Contd…
Infinite Sheet Charge
Contd…
Uniformly Charged Sphere
ρv C/m3.
Contd…
Contd…
Contd…
Contd…
Electric Potential
Contd…
Contd…
Relations Between E and V
Which is the Second of the four Maxwell's equations to be derived.
UNIT-II
Introduction to Electromagnetism | |
| |
Electrostatics
Basic definitions
Coulomb’s Inverse Square Law
Coulomb’s inverse square law gives the force between the two charges. According to this law, the force (F) between two electrostatic point charges (q1 and q2) is proportional to the product of the charges and inversely proportional to the square of the distance (r) separating the charges.
F ∝ q1 q2
(or)
q1
q2
medium
r
F ∝
1
r 2
r
2
F = K
q1 q2
where K is proportionality constant which depends on the nature of the medium.
This force acts along the line joining the charges. For a dielectric medium of relative permittivity εr ,the value
of K is given by,
=
1 1
4πε 0ε r
K =
4π where ε = permittivityεof the medium.
Electric field
0
volt m-1 (or) NC-1
b
Fb
aq
a
4πε ˆra
2
= qa r
E
=
Electrostatic Potential (V)
Potential
r
V = − ∫ E . dx
∞
dx
q
4πε 0 x 2
r
= − ∫
∞
q
4πε 0 r
q
4πε
0
=
⎡1
⎢⎣ r
1 ⎤
∞
⎥⎦
−
V =
Electric lines of force
Properties of electric lines of force
Representation of electric lines of force for Isolated positive and negative charges.
+
q
-q
Electric flux
E
ds
Flux of the
electric field
dφ =E dscos θ= (E cos θ).ds
= (Component of E along the direction of the normal × area)
The flux over the entire surface = φ =
Unit:
Nm2 C − 1
Electric flux expression
The electric flux normal to the
area ds = dφ =
E
ds
∫ dφ
S
= ∫ E cosθ . ds
S
.
⎝ ε0 ⎠
φ = ⎛⎜ 1 ⎞⎟ q
⎝ ε ⎠
0
∫ E ds
cosθ
φ = ⎛⎜ q ⎞⎟ =
(or)
Gauss theorem (or) Gauss law
times the net charge q enclosed by the
to the 1/ε0
surface.
Electric flux density (or) Electric displacement vector (D)
D =
φ
A
Unit : Coulomb / m2
Permittivity (μ)
Permittivity is defined as the ratio of electric displacement vector (D) in a dielectric medium to the applied electric field strength (E).
ε =
D
E
ε = ε 0 ε r
medium
Unit: Farad /metre
Magnetostatics
Magnetic fields.
Magnetic dipole
Magnetic dipole moment (μm)
μm = m x l
μm = i x a
Unit: ampere (metre)2
Magnetic moment
A
μm
i
Unit Area
A
Magnetic flux (φ)
Unit: weber.
Magnetic flux density (or) Magnetic induction (B)
| |||
B = Magnetic Flux weber/m2 (or)Tesla = φ | | B = F = Force experienced m Pole strength | |
| |||
μ μ
⎝ m ⎠
i.e. H = 1 ×⎛ ⎜F ⎞=⎟ B ampereturns / metre
μ = permeability of the medium.
Magnetic field strength (or) Magnetic field intensity (H)
Magnetization (or) Intensity of Magnetization (M)
Magnetic susceptibility (χ)
χ = M
H
(no unit)
Magnetic permeability (μ)
B H
i.e.
μ = μ0 μr
=
⮚
μ r = relative permeability of the medium
sample to the applied magnetic field intensity.
μ0
μ r
=
(μNo unit)
Relation between μr and χ
Total flux density (B) in a solid in the presence of magnetic field can be given as B = μ0 (H+M)
μr = 1 +
Then μr can be related to χ as
χ
Relative permeability (μr)
Bohr Magneton (μB)
.
e h
2m2π
eh
4πm
=
=
unpaired electron in an atom.
It is the fundamental quantum of magnetic moment.
1 Bohr Magneton
1μB = 9.27 x 10−24 ampere metre 2
Current density (J)
dI ds
J =
I =
∫
J.ds
S
Conduction Current Density ( J1)
V = IR (1)
For a length l and potential difference V,
V=El (2)
where E = electric field intensity.
J 1 = σ
E
IR = El
Expression for J1
From equations V =
IR and V= El
(3)
l
A
⎝ σ
⎟ ⎜ ⎟
⎠ ⎝
A ⎠
ρ
= ⎛ ⎜
1
⎞ ⎛ l ⎞
R =
(4)
Using (4) in (3)
⎝ ⎠
I
⎛
⎜ σA
l ⎟⎞ = E.l
I
σA
= E (or)
⎛⎜ I ⎞⎟ =σE
⎝ A ⎠
or
(5)
Displacement Current Density( J 2 )
In a capacitor, the current is given by,
I
c
= dQ = d( CV ) = C. dV dt dt dt
In a parallel plate capacitor, the capacitance is given by,
C =
(1)
(2)
εA d
Using equation (2) in (1)
(or)
dt
.
IC A
ε dV d dt
=
⎟.
⎠
⎜
⎝
d
⎛ εA ⎞
dV
IC =
J2 = Displacement current density =
dE = d (εE)
dt dt
⎣
ε ⎡ d
⎢ dt
⎜ d ⎟⎥
⎝ ⎠⎦
⎛ V ⎞⎤
= ε
dt
J
= d D
2
The net current density = J = J1 + J2
dt
J = σ E
[sinceD = ε E = Electric Displacement vector]
UNIT 3
EM WAVE CHARACTERISTICS
Waves in
both space and time.
In this chapter, our major goal is to solve Maxwell's equations and derive EM wave motion in
the following media:
where ω is the angular frequency of the wave. Case 3, for lossy dielectrics, is the most general case and will be considered first. Once this general case is solved, we simply derive other cases (1,2, and 4) from it as special cases by changing the values of σ,ε, and μ. However, before we consider wave motion in those different media, it is appropriate that we study the characteristics of waves in general. This is important for proper understanding of EM waves.
Wave motion occurs when a disturbance at point A, at time t0, is related to what happens at point B, at time t > t0. A wave equation is a partial differential equation of the second order. In one dimension, a scalar wave equation takes the form of
----(1) where u is the wave velocity.
In general, waves are means of transporGtinegneneerrgaylor information. A wave is a function of
in which the medium is source free (ρv = 0, J = 0). It can be solved by following procedure,
----(2a)
----(2b) or ----(2c)
If we particularly assume harmonic (or sinusoidal) time dependence ejω t, eq. (1) becomes
(3)
Where β= ω /u and Es is the phasor form of E. The solutions to eq. (3) are
----(4a) ----(4b) and
----(4c)
where A and B are real constants.
For the moment, let us consider the solution in eq. (4a). Taking the imaginary part
of this equation, we have
This is a sine wave chosen for simplicity; we have taken the real part of eq. (4a). Note the following characteristics of the wave in eq. (5):
3.(ω t- βz)isthe phase (in radians)ofthe w ave;itdependson tim e t and space variable z. 4.ω is the angular frequency (in radians/second); β is the phase constant or wave number (in radians/meter).
----(5)
Due to the variation of E with both time t and space variable z, we may plot E as a function of t by keeping z constant and vice versa.
The plots of E(z, t = constant) and E(t, z = constant) are shown in Figure 1(a) and (b), respectively.
From Figure 1(a), we observe that the wave takes distance λ to repeat itself and hence λ is called the wavelength (in meters).
From Figure 1(b), the wave takes time T to repeat itself; consequently T is known as the period (in seconds). Since it takes time T for the wave to travel
distance λ atthe speed u, we expect
But T = l/f, where/is the frequency (the number of cycles per second) of the wave in
Hertz (Hz). Hence,
----(6b)
----(6a)
Plot of E(z, t) =A sin (ω t-βz)(a)w ith constantt,(b)with constant z.
----(7b)
and
Because of this fixed relationship between wavelength and frequency, one can identify the
position of a radio station within its band by either the frequency or the wavelength.
Usually the frequency is preferred. Also, because From Eq (6) & (7) We have
----(7a)
----(7c) ----(8)
Equation (8) shows that for every wavelength of distance traveled, a wave undergoes
a
phase change of 2π radians.
We will now show that the wave represented by eq. (5) is traveling with a velocity u in the +z direction. To do this, we consider a fixed point P on the wave. We sketch eq.
(5) at times t = 0, t/4, and t/2 as in Figure 2. From the figure, it is evident that
as the wave advances with time, point P moves along +z direction. Point P is a point of
constant phase, therefore
----(9)
and
In summary, we note the following:
1. A wave is a function of both time and space.
Plot of E(z, t) = A sin(ωt- βz)attim e (a)t = 0, (b) t = T/4,
(c) t = t/2; P moves along +z direction with velocity u.
wave).
4. Since sin (-ψ )= -sin ψ = sin (ψ ± π), Where as cos(- ψ )= cosψ ,
----(10a)
----(10b)
----(10c)
----(10d)
Where ψ = ωt± βzW ith eq.(10),any tim e-harmonic wave can be represented
in the form of sine or cosine.
Wave Propagation in Lossy Dielectrics
A lossy dielectric is a medium in which an EM wave loses power as it propagates
due to poor conduction.
Consider a linear, isotropic, homogeneous, lossy dielectric medium that is charge free
(ρv = 0). Assuming and suppressing the time factor ejω t, Maxwell's equations becomes
----(11) ----(12)
----(13)
----(14)
Taking the curl of both sides of eq. (13) gives
----(15)
Applying the vector identity ----(16) to the left-hand side of eq. (15) and invoking eqs. (11) and (14), we obtain
or
----(17)
Where
----(18)
and γ iscaled the propagation constant(in perm eter)ofthe m edium .By a sim ilar
procedure, it can be shown that for the H field,
----(19)
Equations (17) and (19) are known as homogeneous vector Helmholtz 's equations or
simply vector wave equations.
In Cartesian coordinates, eq. (17), for example, is equivalent to three scalar wave oenqeufaotrioenasc,h component of E along ax , ay , and az. Since γ in eqs.(17)to (19)isa com plex
quantity, we may let
----(20)
We obtain α and β from eqs.(18)and (20)by noting that
----(21)
and
----(22)
From eqs. (21) and (22), we obtain
----(23)
----(24)
Without loss of generality, if we assume that the wave propagates along +az and that Es
has only an x-component, then
----(25)
Substituting this into eq. (17) yields
----(26) Hence
or
----(27)
This is a scalar wave equation, a linear homogeneous differential equation, with solution
----(28)
where Eo and E'o are constants. The fact that the field must be finite at infinity requires that E'o = 0. Alternatively, because eγzdenotes a wave traveling along —az whereas we assume wave propagation along az, E'o = 0. Whichever way we look at it, E'o = 0. Inserting the time factor ejωtinto eq. (28) and using eq. (20), we obtain
----(29)
A sketch of |E| at times t = 0 and t = ∆tis portrayed in Figure , where it is evident that E has only an x-component and it is traveling along the +z direction. Having obtained E(z, t), we obtain H(z, t) either by taking similar steps to solve eq. (19) or by using eq. (29) in conjunction with Maxwell's equations. We will eventually arrive at
----(30)
And η isa com plexquantity know n asthe intrinsic impedance (in ohms) of the medium. It can be shown by following the steps as
----(32)
E-field with x-component traveling along +z direction at times t = 0 and t = ∆t;ar ow sindicate instantaneous values of E.
where
----(31)
----(33)
where 0< θn< 45°. Substituting eqs. (31) and (32) into eq. (30) gives
or
----(34)
Notice from eqs. (29) and (34) that as the wave propagates along az, it decreases or attenuates in amplitude by a factor e-αz, and hence a is known as the attenuation constant or attenuation factor of the medium. It is a measure of the spatial rate of decay of the wave in the medium, measured in nepers per meter (Np/m) or in decibels per meter (dB/m). An attenuation of 1 neper denotes a reduction to e-1 of the original value whereas an increase of 1 neper indicates an increase by a factor of e. Hence, for
voltages
We also notice from eqs. (29) and (34) that E and H are out of phase by θn, at any instant of time due to the complex intrinsic impedance of the medium. Thus at any time, E leads H (or H lags E) by θn. Finally, we notice that the ratio of the magnitude of the conduction current density J to that of the displacement current density Jd in a lossy medium is
----(35)
or ----(36)
where tanθ is known as the loss tangent and θ is the loss angle of the medium as illustrated in Figure. Although a line of demarcation between good conductors and lossy dielectrics is not easy to make, tanθ or θ may be used to determine how lossy a medium is.
A medium is said to be a good (lossless or perfect) dielectric if tan θ is very small
(σ<<ω ε)or a good conductor if tan θ is very large (σ>>ω ε).
From the viewpoint of wave propagation, the characteristic behavior of a medium depends not only on its constitutive parameters σ,ε and μ but also on the frequency of operation.
A medium that is regarded as a good conductor at low frequencies may be a good dielectric at high frequencies.
PLANE WAVES IN LOSSLESS
DIInELaECTRICS
lossless dielectric, σ<<ω ε.
We except
that
----(37)
Substituti ng these into eqs.
(23) and
--(-2-4(3) 9g)ives
----(38a)
----(38b)
Also
Thus E and H are in time phase with each other.
PLANE WAVES IN FREE
SPACE
This is a special case of what we considered
----(40)
----(43b)
Thus we simply replace ε by εo and μ by μo in eq. (38) or we substitute eq. (40) directly into eqs. (23) and (24). Either way, we obtain
----(41a)
----(41b)
where c = 3 X 108 m/s, the speed of light in a vacuum. The fact that EM wave travels in free space at the speed of light is significant. It shows that light is the manifestation of an EM wave. In other words, light is characteristically electromagnetic.
By substituting the constitutive parameters in eq. (40) into eq. (33), θn = 0 and η= η0 where
η0 is called the intrinsic impedance of free space and is given by
----(42) ----(43a) and
The plots of E and H are shown in Figure (a). In general, if aE, aH, and ak are unit vectors
along the E field, the H field, and the direction of wave propagation;
or
(a) Plot of E and H as functions of z at t = 0;
(b) plot of E and H at z = 0. The arrows indicate instantaneous values.
Both E and H fields (or EM waves) are everywhere normal to the direction of wave propagation, ak. That means, the fields lie in a plane that is transverse or orthogonal to the direction of wave propagation. They form an EM wave that has no electric or magnetic field components along the direction of propagation; such a wave is called a transverse electromagnetic (TEM) wave. Each of E and H is called a uniform plane wave because E (or H) has the same magnitude throughout any transverse plane, defined by z = constant. The direction in which the electric field points is the polarization of a TEM wave. The wave in eq. (29), for example, is polarized
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or
----(44)
PLANE WAVES IN GOOD
This is another special case. A perfect, oCr gOoNodDcUonCdTucOtoRr,Sis one in which σ>>ωεso that
σ/ωε→ ∞ thatis,
Hence, eqs. (23) and (24) become
----(45)
----(46a)
----(46b)
Also
and thus E leads H by 45°. If
----(47)
----(48a)
then
----(48b)
Therefore, as E (or H) wave travels in a conducting medium, its amplitude is attenuated by the factor e-αz.
Illustration of skin depth.
The distance δ,through w hich the w ave am plitude decreases by a factor e-1 (about 37%) is
called skin depth or penetration depth of the medium; that is
The skin depth is a measure of the depth to which an EM wave can penetrate the medium.
or
----(49a)
For good conductors eqs. (46a) and (49a) give
----(49b)
Poynting
This theorem state thaTt htheetootarlecommplex power fed in to volume is
en as follows
equal to the algebraic sum of the active power dissipated as heat plus the reactive power proportional to the difference between time average magnetic & electric energies stored in the volume, plus the complex power transmitted across the surface enclosed by the volume.
The time average of any two complex vectors is equal to the real part of the product of one complex vector multiplied by the complex conjugate of the other vector
The time average of the instantaneous Poynting vector in steady form is given by
(1)
Where stands for average ½
represents complex power when peak values are used, & asterisk indicates complex conjugate.
The complex Poynting Vector is defined as ------(2) Maxwell’s
Equations in Frequency domain are giv
(3)
-----(4)
LHS of Eq (7) is
b
y vector identity. So we have
(8)
(
---
Substituting Eq(2) & (5) in Eq (8) we have
9) Integrating the above equation over the volume and applying Gauss Theorem to the last term on RHS gives
-(10)
The above equation is known as Complex Poynting Theorem, or Poynting theorem in frequency domain.
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(5)
(6)
Subtracting (5) from (6) results as
(7)
Where E.E* is replaced by | E |2 & H.H* is replaced by | H |2
Further
The above Equation is simplified as follows
Uniform Plane Wave
Reflection of Uniform Plane wRaveeisfbrloeadlcy ctaisonin to two ways
Normal Incidence
The simplest case of reflection is normal incidence it is represented in the following figure. In medium 1 the fields are the sum of incident & reflected waves. So,
(1)
(2)
In medium 2 there are only Transmitted waves
(3)
----(4)
Oblique Incidence
E is in the Plane of Incidence (Parallel polarization)
When ever a wave is incident obliquely on the boundary surface between media, the polarization of the wave is vertical or horizontal if the electric field is normal or parallel to boundary surface.
Figure shows a loss less dielectric medium. The phase constant of the two media in x direction on the interface are equal as required by the continuity of tangential E & H boundary
We have
---(5)
---(6)
H is in the Plane of Incidence (Perpendicular polarization)
If H is in the plane of incidence the components of H are
The components of Electric intensity E normal to plane of incidence are
The wave impedance in Z direction is given by
UNIT 4 & 5
Transmission Lines I & II
TRANSMISSION LINES
In an electronic system, the delivery of power requires the connection of two wires between the source and the load. At low frequencies, power is considered to be delivered to the load through the wire.
In the microwave frequency region, power is considered to be in electric and magnetic fields that are guided from place to place by some physical structure. Any physical
structure that will guide an electromagnetic wave place to place is called a
Transmission Line.
Types of Transmission Lines
Transmission lines are commonly used in power distribution (at low frequencies) and in communications (at high frequencies). Various kinds of transmission lines such as the twisted-pair and coaxial cables (thinnet and thicknet) are used in computer networks such as the Ethernet and Internet.
A transmission line basically consists of two or more parallel conductors used to connect a source to a load. The source may be a hydroelectric generator, a transmitter, or an oscillator; the load may be a factory, an antenna, or an oscilloscope, respectively. Typical transmission lines include coaxial cable, a two-wire line, a parallel-plate or planar line, a wire above the conducting plane, and a microstrip line.
Figure: Cross-sectional view of typical transmission lines: (a) coaxial line, (b) two-wire line, (c) planar line, (d) wire above conducting plane, (e) microstrip line.
At low frequencies, the circuit elements are
lumped since voltage and current waves affect the entire circuit at the same time.
At microwave frequencies, such treatment of circuit elements is not possible since voltage and current waves do not affect the entire circuit at the same time.
The circuit must be broken down into unit sections within which the circuit elements are considered to be lumped. This is because the dimensions of the circuit are comparable to the wavelength of the waves according to
the formula:
where,
c = velocity of light
λ = c/f
f = frequency of voltage/current
Electrical Dimensions, Circuit and Field Analysis
TRANSMISSION LINE PARAMETERS
Distributed Line Parameters at High Frequencies
Figure: Common transmission lines: (a) coaxial line, (b) two-wire line, (c) planar line.
1. The line parameters R, L, G, and C are not discrete or lumped but distributed as shown in Figure. By this we mean that the parameters are uniformly distributed along the entire length of the line.
3. For each line, the conductors are characterized by σc µc εc= εo, and the homogeneous
dielectric separating the conductors is characterized by σ,µ, ε.
2.For Each Line
LC = με
and
electromagnetic (TEM) wave propagating along the line. This wave is a non uniform plane wave and by means of it, power is transmitted through the line.
The Poynting vector (E X H) points along the transmission line. Thus, closing the switch simply establishes a disturbance, which appears as a transverse
let us consider how an EM wave propagates through a two-conductor transmission line. For example, consider the coaxial line connecting the generator or source to the load as in Figure (a) . When switch S is closed, the inner
conductor is made positive with respect
to the outer one so that the E field is radially outward as in Figure (b). According to Ampere's law, the H field
encircles the current carrying conductor Figure (b).
(a) Coaxial line connecting the generator to the load
as in
(b) E and H fields on the coaxial line.
TRANSMISSION LINE EQUATIONS (lossy
For conductors(σc ≠t∞yp),eF)or dielectric (σ≠ 0).
uniquely related to voltage V and
As mentioned, a two-conductor transmission line supports a TEM wave; that is, the electric and magnetic fields on the line are transverse to the direction of wave propagation. An important property of TEM waves is that the fields E and H are
current I, respectively:
(1)
Let us examine an incremental portion of length Δz of a two-conductor transmission line.
We intend to find an equivalent circuit for this line and derive the line equations.
We expect the equivalent circuit of a portion of the line to be as in Figure above. The model in Figure is in terms of the line parameters R, L, G, and C, and may represent any of the two-conductor lines. The model is called the L-type equivalent circuit; there are other possible types. In the model of Figure, we assume that the wave propagates along the +z- direction, from the generator to the load.
By applying Kirchhoff's voltage law to the outer loop of the circuit in Figure,
Taking the limit of eq. (2) as Δz--> 0 leads to
we obtain
or
------- (2)
(3)
Similarly, applying Kirchhoff's current law to the main node of the circuit in Figure gives
or
(4)
As Δz—> 0, eq. (4) becomes
(5)
If we assume harmonic time dependence so that
(6a)
(6b) where Vs(z) and Is(z) are the phasor forms of V(z, t) and I(z, t), respectively, eqs. (3) and
(5) become
(7)
(8)
In the differential eqs. (7) and (8), Vs and Is are coupled. To separate them, we take
the second derivative of Vs in eq. (7) and employ eq. (8) so that we obtain
or
------- (9)
Where
Similarly
(10)
------- (11)
We notice that eqs. (9) and (11) are, respectively, the wave equations for voltage and current similar in form to the wave equations obtained for plane waves. Thus, in our usual notations, ϒ in eq. (10) is the propagation constant , α is the attenuation constant (in nepers per meter or decibels2 per meter), and β is the phase constant (in radians per meter). The wavelength λ and wave velocity u are, respectively, given by
(12)
(13)
The solutions of the linear homogeneous differential equations (9) and (11) are
(14)
(15)
o o o
where Vo +, V -, I +, and I - are wave amplitudes; the + and — signs, respectively, denote
wave traveling along +z and –z directions, as is also indicated by the arrows. Thus, we
obtain the instantaneous expression for voltage as
(16)
Characteristic impedance of a Transmission Line (Zo )
The characteristic impedance Zo of the line is the ratio of positively traveling voltage wave to current wave at any point on the line or negatively traveling voltage wave to current wave at any point on the line .
Zo is analogous to η,the intrinsic impedance of the medium of wave propagation. By
substituting eqs. (14) and (15) into eqs. (7) and (8) and equating coefficients of terms e
ϒz
and e -ϒz, we obtain
(17)
(18)
where Ro and Xo are the real and imaginary parts of Zo. The propagation constant ϒ and the characteristic impedance Zo are important properties of the line because they both depend on the line parameters R, L, G, and C and the frequency of operation. The reciprocal of Zo is the characteristic
www.android.previousquestionpapers.com | www.previousquestionapadpmers.ictotma|nhcttpes:/Y/toel,egtrhama.tmei/sjn,tuYao = 1/ZO.
Lossless Line (R = 0 = G)
A transmission line is said lo be lossless if the conductors of the line are perfect
(σc =∞ )and the dielectric medium separating them is lossless (σ= 0).
R = 0 = G ------- (19)
This is a necessary condition for a line to be lossless. Thus for such a line, eq.
(19) forces eqs. (10), (13), and (18) to become
------- (20a)
------- (20b)
------- (20c)
Distortion less Line (R/L = G/C)
A signal normally consists of a band of frequencies; wave amplitudes of different frequency components will be attenuated differently in a lossy line as α is frequency dependent. This results in distortion.
A distortion less line is one in which the attenuation constant α is frequency independent while the phase constant β is linearly dependent on frequency. From the general expression for α and β [in eq. (10)], a distortion less line results if the line parameters are such that
R/L = G/C ------- (21)
Thus, for a distortion less line,
or
------- (22a)
------- (22c)
Note that
Also we have
------- (22b)
Transmission Line Characteristics
NOTE:
For our analysis, we shall restrict our discussion to lossless transmission lines.
INPUT IMPEDANCE, SWR, AND
o
Consider a transmission linPeOof WlengEthR, characterized by ϒand Z , connected
to a load ZL as shown in Figure. Looking into the line, the generator sees the line with the load as an input impedance Zin. It is our intention to determine the input impedance, the standing wave ratio (SWR), and the power flow on the line.
Let the transmission line extend from z = 0 at the generator to z = at the
load. First of all, we need the voltage and current waves in eqs. (14) and (15), that is
(23)
(24)
where eq. (17) has been incorporated. To find Vo - and Vo +, the terminal conditions must be given.
Ξ
Figure (a) Input impedance due to a line terminated by a load;
(b) equivalent circuit for finding Vo and Io in terms of Zin at the input.
If we are given the conditions at the input,
------- (25)
substituting these into eqs. (23) and (24) results in
------- (26a)
------- (26b)
If the input impedance at the input terminals is Zin, the input voltage Vo and the input
current Io are easily obtained from Figure (b) as
(27)
On the other hand, if we are given the conditions at the load, say
(28)
Substituting these into eqs. (23) and (24) gives
------- (29a)
------- (29b)
Next, we determine the input impedance Zin = Vs(z) / Is(z) at any point on the line. At
the generator, for example, eqs. (23) and (24) yield
(30)
Substituting eq. (29) into (30) and utilizing the fact that
or
(lossy) ------- (31)
we get ------- (32)
Although eq. (32) has been derived for the input impedance Zin at the generation end, it is a general expression for finding Zin at any point on the line.
For a lossless line, y = jβ,tanh jβ = jtanβ ,and Zo = Ro, so eq. (32) becomes
(lossless)
------- (33)
showing that the input impedance varies periodically with distance from the load. The Quantity β in eq. (33) is usually referred to as the electrical length of the line and can be expressed in degrees or radians.
We now define ГL as the voltage reflection coefficient (at the load). ГL is the ratio of
the voltage reflection wave to the incident wave at the load, that is,
(34)
Substituting Vo - and Vo + m eq. (29) into eq. (34) and incorporating VL = ZL IL gives
But z = - '. Substituting and combining with eq. (34), we get
(36)
The current reflection coefficient at any point on the line is negative of the voltage reflection coefficient at that point.
Thus, the current reflection coefficient at the load is
we define the standing wave ratio s (SWR) as
(37)
= ------- (35)
The voltage reflection coefficient at any point on the line is the ratio of the magnitude of the reflected voltage wave to that of the incident wave.
and
----- (38b)
----- (38a)
It is easy to show that Imax = Vmax/Zo and Imin = Vmin/Zo. The input impedance Zin in eq. (33) has maxima and minima that occur, respectively, at the maxima and minima of the voltage and current standing wave. It can also be shown that
As a way of demonstrating these concepts, consider a lossless line with characteristic impedance of Zo = 50Ω. For the sake of simplicity, we assume that the line is terminated in a pure resistive load ZL = 100Ω and the voltage at the load is 100 V (rms). The conditions on the line are displayed in
Figure . Note
conditions on
from the figure that the line repeat
themselves every half wavelength.
Voltage and current wave patterns on a lossless line terminated by a resistive load.
A transmission line is used in transferring power from the source to the load. The average input power at a distance from the load is given by
where the factor 1/2 is needed since we are dealing with the peak values instead of the rms values. Assuming a lossless line, we substitute eqs. (23) and (24) to obtain
Since the last two terms are purely imaginary, we have
----- (39)
The first term is the incident power Pi , while the second term is the reflected power Pr. Thus eq. (39) may be written as Pt = Pi — Pr
where Pt is the input or transmitted power and the negative sign is due to the negative going wave since we take the reference direction as that of the voltage/current traveling toward the right. We should notice from eq. (39) that the power is constant and does not depend on since it is a lossless line. Also, we should notice that maximum power is delivered to the load when ϒ= 0, as expected.
pure reactance, which could be the value of . The variation of
We notice from eq. (40) that Zin is a capacitive or inductive depending on Zin with is shown in Figure (a).
We now consider special cases when the line is connected to load ZL = 0, ZL =∞ ,
and ZL = Zo. These special cases can easily be derived from the general case.
Shorted Line (ZL = 0)
For this case, eq. (33) becomes
----- (40a)
Also
----- (40b)
Open-Circuited Line (ZL =∞ )
In this case, eq. (33) becomes
----- (41a)
and
----- (41b)
The variation of Zin with is shown in Figure (b). Notice from eqs. (40a) and (41a) that
----- (42)
Matched Line (ZL = Zo)
This is the most desired case from the practical point of view. For this case, eq. (33) reduces to
----- (43a) and ----- (43b)
that is, Vo = 0, the whole wave is transmitted and there is no reflection. The incident power is fully absorbed by the load. Thus maximum power transfer is possible when a transmission line is matched to the load.
THE SMITH
Prior to the advent of digitaCl cHomApuRterTs and calculators, engineers developed all
sorts of aids (tables, charts, graphs, etc.) to facilitate their calculations for design and analysis.
To reduce the tedious manipulations involved in calculating the characteristics of transmission lines, graphical means have been developed. The Smith chart is the most commonly used of the graphical techniques. It is basically a graphical indication of the impedance of a transmission line as one moves along the line. It becomes easy to use after a small amount of experience. We will first examine how the Smith chart is constructed and later employ it in our calculations of transmission line characteristics such as ΓL, s, and Zin. We will assume that the transmission line to which the Smith chart will be applied is lossless (Zo = Ro) although this is not fundamentally required.
The Smith chart is constructed within a circle of unit radius (| Γ|≤ 1)asshow n in
Figure. The construction of the chart is based on the relation in eq. 35 that is,
=
----- (44)
or
----- (45)
where Γr and Γi, are the real and imaginary parts of the reflection coefficient Γ.
Instead of having separate Smith charts for transmission lines with different characteristic impedances such as Zo = 60,100, and 120 Ω, we prefer to have just one that can be used for any line. We achieve this by using a normalized chart in which all impedances are normalized with respect to the characteristic impedance Zo of the particular line under consideration. For the load impedance ZL, for example, the normalized impedance ZL is
given by
----- (46)
Substituting eq. (46) into eqs. (44) and (45) gives
----- (47a)
or
----- (47b)
Normalizing and equating components, we obtain
Rearranging terms in eq. (48) leads to
----- (48a)
----- (48b)
----- (49)
and
----- (50)
Each of eqs. (49) and (50) is similar to
----- (51)
which is the general equation of a circle of radius a, centered at (h, k). Thus eq. (49) is an
r-circle (resistance circle) with
----- (52a)
----- (52b)
For typical values of the normalized resistance r, the corresponding centers and radii of the r-circles are presented in Table. Typical examples of the r-circles based on the data in Table are shown in Figure. Similarly, eq. (50) is an x-circle (reactance circle) with
----- (53b)
----- (53a)
Typical r-circles for r = 0,0.5, 1,2, 5, ∞
Table below presents centers and radii of the x-circles for typical values of x, and Figure below shows the corresponding plots. Notice that while r is always positive, x can be positive (for inductive impedance) or negative (for capacitive impedance).
If we superpose the r-circles and x-circles, what we have is the Smith chart shown in Figure in next slide on the chart, we locate a normalized impedance z = 2 + j , for example, as the point of intersection of the r = 2 circle and the x = 1 circle. This is point P1 in Figure. Similarly, z = 1 - j0.5 is located at P2, where the r = 1 circle and the x = -0.5 circle intersect.
Typical x circles for x = 0, ± 1/2, ±1, ±2, ±5, ±∞ .
Apart from the r- and x-circles (shown on the Smith chart), we can draw the S-circles or constant standing-wave-ratio circles (always not shown on the Smith chart), which are centered at the origin with s varying from 1 to ∞ . The value of the standing wave ratio s is
determined by locating where an s-circle crosses the Гr axis. Typical examples of S circles for S = 1,2, 3, and ∞ are shown in Figure. Since
|Г| and S are related according to eq. (37), the S circles are sometimes referred to as |Г|-circles with |Г| varying linearly from 0
to 1 as we move away from the center O toward the periphery of the chart while s varies nonlinearly from 1 to ∞ .
The following points should be noted about the
Smith chart:
1. At point Psc on the chart r = 0, x = 0; that is, ZL = 0 + j0 showing that Psc represents a short circuit on the transmission line. At point Poc, r = ∞ and x = ∞ or ZL = ∞ +j∞ , which implies that Poc corresponds to an open circuit on the line. Also at Poc, r = 0 and x = 0, showing that Poc is another location of a short circuit on the line.
2. A complete revolution (360°) around the Smith chart represents a distance of λ/2 on the line. Clockwise movement on the chart is regarded as moving toward the generator (or away from the load) as shown by the arrow G in Figure (a) and (b). Similarly, counterclockwise movement on the chart corresponds to moving toward the load (or away from the generator) as indicated by the arrow L in Figure. Notice from Figure that at the load, moving toward the load does not make sense (because we are already at the load). The same can be said of the
case when we are at the generator end.
(a) Smith chart illustrating scales around the
periphery and movements around the chart,
(a)
(b)
3. There are three scales around the periphery of the Smith chart as illustrated in Figure. The three scales are included for the sake of convenience but they are actually meant to serve the same purpose; one scale should be sufficient. The scales are used in determining the distance from the load or generator in degrees or
distance on the line from the generator end in terms
wavelengths. The outermost scale is used to determine the
of
wavelengths, and the next scale determines the distance from the load end in terms of wavelengths. The innermost scale is a protractor (in degrees) and is primarily used in determining θГ; it can also be used to determine the distance from the load or generator. Since a λ/2 distance on the line corresponds to a
movement of
360° on the chart, A distance on the line
corresponds to a 720° movement on the chart. λ→ 720o
Thus we may ignore the other outer scales and use the protractor (the innermost scale) for all our θГ and distance calculations.
Based on these important properties, the Smith chart may be used to determine, among other things, (a) Г = |Г| LθГ and s;
(b) Zin or Yin; and (c) the locations of Vmax and Vmin provided that we are given Zo, ZL, and the length of the line. Some examples will clearly show how we can do all these and much more with the aid of the Smith chart, a compass, and a plain straightedge.
TRANSIENTS ON TRANSMISSION LINES
In circuit analysis, when a pulse generator or battery connected to a
transmission line is switched on, it takes some time for the current and voltage on the line to reach steady values. This transitional period is called the transient. The transient behavior just after closing the switch (or due to lightning strokes) is usually analyzed in the frequency domain using Laplace transform. For the sake of
o
Suppose that the line is driven by a pulse generator of voltage Vg with internal impedance Zg at z = 0 and terminated with a purely resistive load ZL. At the instant t = 0 that the switch is closed, the starting current "sees" only Zg and Zo, so the initial situation can be described by the equivalent circuit of Figure (b). From the figure, the starting current at
z = 0, t = 0+ is given by
Figure :Transients on a transmission line: (a) a line driven by a pulse generator, (b) the equivalent circuit at z = 0, t = 0+.
---(1)
and the initial voltage is ---(2)
After the switch is closed, waves I+ = Io and V+ = Vo propagate toward the load at the speed
---(3)
Since this speed is finite, it takes some time for the positively traveling waves to reach the load and interact with it. The presence of the load has no effect on the waves before the transit time given by
---(4)
After t1 seconds, the waves reach the load. The voltage (or current) at the load is the sum
of the incident and reflected voltages (or currents). Thus
---(5)
and
---(6)
where ГL is the load reflection coefficient given that is,
The reflected waves V -= ГL Vo and I-= — ГL Io travel back toward the generator in addition
ady on the line. At time t = 2t1 the reflected wave
reatcohtehdetwheavgeesnVeoraatnodr, Isooalre
s h--a-v(7e)
or
and
or
where ГG is the generator reflection coefficient given by
Again the reflected waves (from the generator end) V+ = ГG ГL VO and I+ = ГG ГL IO propagate toward the load and the process continues until the energy of the pulse is actually absorbed by the resistors Zg and ZL.
Instead of tracing the voltage and current waves back and forth, it is easier to keep track of the reflections using a bounce diagram, otherwise known as a lattice diagram. The bounce diagram consists of a zigzag line indicating the position of the voltage (or current) wave with respect to the generator end as shown in Figure. On the bounce diagram, the voltage (or current) at any time may be determined by adding those values that appear on the diagram above that time.
Figure : Bounce diagram for (a) a voltage wave, and (b) a current wave.