Eem Notes Chapter1 002

Raghuvir Tomar Electromagnetic Theory Notes For Chapter 1 (Fundamentals) Revision 002

Authors Date Place

Prepared by: Raghuvir Tomar 24th January 2006

LNMIIT Jaipur, India

Reviewed by:

Reviewed by:

EM Theory Notes for Chapter 1 by R.Tomar Rev 002 ______________________________________________________________________

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Revision History Revision Date Description

001 23rd January 2006 Initial draft

002 24th January 2006 Some typos corrected


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Table of Contents

1 FUNDAMENTAL CONCEPTS AND DEFINITIONS .......................................1

1.1 A typical modern communication system ..........................................................1

1.2 Force.......................................................................................................................1

1.3 Energy ....................................................................................................................1

1.4 Power......................................................................................................................1

1.5 Frequency ..............................................................................................................2

1.6 Electrical Current ..................................................................................................2

1.7 Electrical Charge...................................................................................................3

1.8 Electrical Resistance ............................................................................................3

1.9 Electrical Conductance.........................................................................................3

1.10 Electrical Resistivity ..........................................................................................3

1.11 Electrical Conductivity ......................................................................................3

1.12 Electromotive Force (EMF) ...............................................................................3

1.13 Electrical Potential Difference or Voltage........................................................4

1.14 Magnetomotive Force (MMF) ............................................................................4

1.15 Electric Field Strength.......................................................................................4

1.16 Electric Flux Density..........................................................................................4

1.17 Capacitance........................................................................................................4

1.18 Inductance ..........................................................................................................4

1.19 Magnetic Flux Density .......................................................................................4

1.20 Magnetic Field Strength ....................................................................................5

1.21 Permeability........................................................................................................5

1.22 Permittivity .........................................................................................................5

1.23 Network parameters .............................................................................................6 1.23.1 Two-port networks ............................................................................................6

1.23.1.1 Admittance (Y) parameters........................................................................6


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1.23.1.2 Impedance (Z) parameters .......................................................................7 1.23.1.3 Hybrid (h) parameters...............................................................................8 1.23.1.4 ABCD parameters (transmission parameters)...........................................8 1.23.1.5 Scattering (S) parameters .........................................................................9 1.23.1.6 Inter-relationship among network parameters ........................................10

1.23.2 Single-port networks.................................................................................10


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1 FUNDAMENTAL CONCEPTS AND DEFINITIONS

The objective of this chapter is to re-capitulate the fundamental knowledge that would serve as the background for this course.

1.1 A typical modern communication system

A typical modern communication system is shown, in its most basic form, in Fig 1.1. It consists of the transmitter, the propagation medium, and the receiver. There are two different approaches to analyzing a communication system:

1. Circuit analysis (also referred to as the circuit theory) 2. Field analysis (also referred to as the electromagnetic theory)

Circuit analysis is based on concepts like voltage (V) and current (I) and uses well-known relationships like Ohm’s law and Kirchhoff’s law. This method is sufficiently accurate for DC and low frequencies. Electromagnetic theory (also abbreviated as EM theory) is based on concepts like electric field (vector E) and magnetic field (vector H). This approach is suited for higher frequencies as well as for understanding radiation effects. Now we shall briefly review some terminology that will be frequently used in this course.

1.2 Force

Force, F, is a measure of the effort put into doing a job or in causing a physical change. One common unit used for force is newton. One newton is defined as the force required to accelerate an object of 1Kg. mass at the rate of 1m/sec2. Another unit used for force is dyne. One dyne is defined as the force required to accelerate an object of 1 gram mass at the rate of 1cm/sec2.

1.3 Energy

Energy, E, can be defined as force multiplied by distance. One common unit of energy is joule. One joule is defined as the energy spent when a force of 1 newton is exerted through a distance of 1meter. Energy can also be measured in erg. One erg is equal to the energy spent when a force of 1 dyne is exerted through a distance of 1 cm.

1.4 Power

Power is not a concept applicable exclusively to electrical or electronic circuits. The fundamental definition of power is energy per unit time. In case of electrical and


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electronic circuits, power is measured either in Watts (W), or in decibels (dB,dBm,or dBW). The fundamental definitions for these quantities follow. 1 W = 1 Joule/Second (1.1) For electrical engineering purposes, the following definitions are more relevant. 1W = (1 Volt) X( 1 Ampere) (1.2) dB=10log10(P/Pref) (1.3) where P is the unknown power and Pref is a reference power against which P is measured. When Pref=1mW, we define dBm=10log10(P/1mW) (1.4) When Pref=1W, we define dBW=10log10(P/1W) (1.5) It is easy to show that, for the same value of P, dBm=30+dBW (1.6) The term ‘average power’ is the most popularly used one, in specifying radio frequency (RF) and microwave components. The terms ‘pulse power’, ‘peak power’, and ‘peak envelope power (PEP)’ are more specific to radar applications, and to systems using complex modulation schemes like Quadrature Amplitude Modulation (QAM) Quadrature Phase Shift Keying (QPSK), etc.

1.5 Frequency

Frequency is not a concept applicable exclusively to electrical or electronic circuits. The frequency, f, of a repetitive phenomenon is defined as the number of cycles of that repetitive phenomenon per unit time. For electrical and electronic circuits, the word ‘phenomenon’ in the above paragraph is replaced by the word ‘signal’ and the unit of time is usually selected to be a second. We thus define the frequency, f, of a repetitive electrical or electronic signal as f=N/T cycles per second or Hz (1.7) where N is the number of cycles of that repetitive signal, over a period of T seconds.

1.6 Electrical Current

Electrical current, I or A, is the result of net flow or movement of electric charges from one point to another, or across a boundary. Current is usually measured in Amperes. An


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Ampere is defined as the constant current which, if maintained in two straight electrical conductors of infinite length and negligible cross-section, and placed 1 meter apart in vacuum, shall produce a force of 2x10-7 newton/meter between these two conductors.

1.7 Electrical Charge

Electrical charge, Q or q, is a fundamental electrical property that allows a charged particle to attract or repel other particles. Charge can also be thought of as current multiplied by time. The most commonly used unit of charge is coulomb. One coulomb is equal to 1 Ampere multiplied by 1 second.

1.8 Electrical Resistance

Electrical resistance, R, is a measure of the degree to which an object opposes the passage of an electrical current. The unit of resistance is ohm. If 1Watt of power is dissipated in a resistance when 1A current flows through it, the value of the resistance is deemed to be 1 ohm.

1.9 Electrical Conductance

Electrical conductance, G, is the reciprocal of the resistance, R. The unit of G is mho (or siemens).

1.10 Electrical Resistivity

Electrical resistivity ‘rho’ of a material is defined as Rho=(R*A)/L (1.8) Where R is the electrical resistance of a uniform specimen of the material (in ohm), L is the length of the specimen (in meter), and A is the cross-sectional area of the specimen (in m2). The unit of resistivity is ohm-m.

1.11 Electrical Conductivity

Electrical conductivity, ‘sigma’, of a material is the mathematical reciprocal of its electrical resistivity. The unit of conductivity is mho/m.

1.12 Electromotive Force (EMF)

EMF is the measure of the strength of a source of electrical energy. It is measured in volts.


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1.13 Electrical Potential Difference or Voltage

Electrical potential difference (also called voltage), V, between two points is the energy that would be required to move a positive unit electrical charge from one point to another. Potential difference is normally measured in volts. 1 volt=1 joule/1 coulomb

1.14 Magnetomotive Force (MMF)

MMF is the measure of the strength of a source of magnetic energy. It is measured in Amperes.

1.15 Electric Field Strength

Electric field strength, vector E, also called electric field intensity, at any point in a medium is the force experienced by a unit positive charge at that point. The unit of E is newton/coulomb.

1.16 Electric Flux Density

Electric flux density, vector D, at any point in a medium is the total electric charge passing through a surface of area equal to one square meter at that point. The unit of D is coulomb/m2. Electric flux density is also called displacement density.

1.17 Capacitance

Capacitance, C, can be defined as the ratio of total charge, Q, on a charged isolated conductor and the electric potential, V, that the charge has generated on the conductor. Capacitance is measured in farads. One farad is equal to 1 coulomb divided by 1 volt.

1.18 Inductance

Inductance, L, is the property of a circuit or component by virtue of which an emf is induced in it as result of changing magnetic flux. Inductance is measured in Henry. One Henry is equal to 1 volt divided by (1 Ampere/second).

1.19 Magnetic Flux Density

The voltage between the terminals of a loop of wire depends on the rate of change of magnetic flux, ‘phi’, through any surface enclosed by the loop. The unit of ‘phi’ is Weber. ! Weber is equal to 1Volt-Second. The magnetic flux density, vector B, essentially speaking, is the response of a medium or material to the presence of a magnetic field. Mathematically, we can write, B=F/(I*L) (1.9)


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Where B is the magnitude of the flux density, and F is the force experienced by a wire of length L carrying a current I.

1.20 Magnetic Field Strength

Magnetic field strength, also referred to as the magnetic field intensity, is abbreviated by the symbol ‘vector H’. Magnetic field strength between two parallel plane sheets carrying equal and oppositely directed currents is equal to the current per meter width flowing in the sheets. The unit of H is Amperes/meter.

1.21 Permeability

Permeability, ‘µ’, relates the magnetic flux density vector B, and the magnetic field intensity vector H, by means of the following equation.

B=µ *H (1.10) We can also write

µ µ µ µ =µ µ µ µ 0µ µ µ µ r (1.11)

Where µ r is the relative permeability of the medium and µ 0 is the permeability of free-space.

µ 0=4π*10-7 Henry/m (1.12)

1.22 Permittivity

Permittivity, ‘ε’, relates the electrical flux density vector D, and the electric field intensity vector E, by means of the following equation. D=ε*E (1.13) We can also write ε=ε0*εr (1.14) Where εr is the relative permittivity of the medium and epsilon0 is the permittivity of free-space. ε0=(1/(36*π))*10-9 Farad/m (1.15)


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1.23 Network parameters

The electronic networks are broadly divided into the following five categories:

1. Two-port networks. 2. One-port networks. 3. Multiple-port networks. 4. Multi-stage networks. 5. Non-linear networks.

We shall discuss only linear two-port and single-port networks since others are not too relevant in this course.

1.23.1 Two-port networks A two-port network is shown in Fig 1.2, with the definitions of various quantities shown therein. The input-output behavior of such a two-port network can be defined by any of the following five parameters: Y, Z, H, ABCD, and S. The definitions and brief discussions of these parameters follow.

1.23.1.1 Admittance (Y) parameters The admittance or Y parameters relate the input and output currents I1 and I2 to the input and output voltages V1 and V2. Mathematically speaking,

2121111 VYVYI += (1.16) 2221212 VYVYI += (1.17)

Eqs. 1.16 and 1.17 lead to the following definitions for Y11,Y12,Y21,and Y22:

1

111

V

IY =

V2=0 (1.18)

2

112

V

IY =

V1=0 (1.19)

1

221

V

IY =

V2=0 (1.20)

2

222

V

IY =

V1=0 (1.21)

Y parameters, as is clear in the above definitions, can be measured by creating short-circuits at input and output ports (V1=0 or V2=0). The preferred use of Y parameters is below 1MHz where the definitions of voltage, current, and short-circuits are quite unambiguous. An additional factor that discourages the use of Y parameters above


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1MHz is the possibility that, for active devices, unwanted oscillations may occur when the circuit is terminated in a short-circuit.

1.23.1.2 Impedance (Z) parameters The impedance or Z parameters are meant to relate the input and output voltages V1 and V2 to the input and output currents I1 and I2 and, like Y parameters, are preferred for frequencies below 1MHz. Mathematically, the Z parameters are defined by the following two equations.

2121111 IZIZV += (1.22) 2221212 IZIZV += (1.23)

Eqs. 1.22 and 1.23 lead to the following definitions for Z11,Z12,Z21,and Z22:

1

111

I

VZ = I2=0 (1.24)

2

112

I

VZ = I1=0 (1.25)

1

221

I

VZ =

I2=0 (1.26)

2

222

I

VZ =

I1=0 (1.27)

Z parameters can be measured by creating open-circuits at input and output ports (I1=0 or I2=0) as is evident in the above definitions. The preferred use of Z parameters is below 1MHz where the definitions of voltage, current, and open-circuits are quite unambiguous. An additional factor that discourages the use of Z parameters above 1MHz is the possibility that, for active devicess, unwanted oscillations may occur when the circuit is terminated in an open-circuit.


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1.23.1.3 Hybrid (h) parameters The hybrid or h parameters relate input voltage V1 and output current I2 to input current I1 and output voltage V2. Mathematically speaking, we write

2121111 VhIhV += (1.28) 2221212 VhIhI += (1.29)

which lead to the following definitions for h11,h12,h21,and h22.

1

111

I

Vh =

I2=0 (1.30)

2

112

V

Vh =

I1=0 (1.31)

1

121

I

Ih =

V2=0 (1.32)

2

222

V

Ih =

I1=0 (1.33)

h parameters can be measured by creating open-circuits and short-circuits at input and output ports (I1=0 or V2=0). The preferred use of h parameters is below 1MHz where the definitions of voltage, current, open-circuits, and short-circuits are quite unambiguous. An additional factor that discourages the use of h parameters above 1MHz is the possibility that, for active devicess, unwanted oscillations may occur when the circuit is terminated in an open-circuit or in a short-circuit.

1.23.1.4 ABCD parameters (transmission parameters) ABCD parameters express input port voltage V1 and input current I1 as functions of output port voltage V2 and output current I2. These parameters are defined by means of the following two equations.

221 BIAVV −= (1.34) 221 DICVI −= (1.35)

Eqs. (1.34) and (1.35) easily lead to the following definitions for ABCD parameters.

2

1

V

VA =

I2=0 (1.36)


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2

1

I

VB −=

V2=0 (1.37)

2

1

V

IC =

I2=0 (1.38)

2

1

V

ID −=

I2=0 (1.39)

ABCD parameters, like h parameters, are hybrid in nature. Their biggest advantage is their easy algebraic manipulation for multi-stage networks, since the ABCD matrix is directly multiplicable for cascaded networks. ABCD parameters can be measured by creating open-circuit (I2=0 ) and short-circuit (V2=0) at the output port. The preferred use of ABCD parameters is below 1MHz where the definitions of voltage, current, open-circuits, and short-circuits are quite unambiguous. An additional factor that discourages the use of ABCD parameters above 1MHz is the possibility that, for active circuits, unwanted oscillations may occur when the circuit is terminated in an open-circuit or in a short-circuit.

1.23.1.5 Scattering (S) parameters Scattering or S-parameters are based on incident and reflected powers and are the most unambigiously-defined parameters for RF and microwave frequencies. These parameters are defined by means of the following two equations.

2121111 aSaSb += (1.40)

2221212 aSaSb += (1.41) where a1 and a2 are quantities representing the two incident signals and b1 and b2 are quantities representing the two reflected signals. The definitions of S parameters, derived from eqs. (1.40) and (1.41), follow.

1

1

11

a

bS =

a2=0 (1.42)

2

1

12

a

bS =

a1=0 (1.43)


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1

2

21

a

bS =

a2=0 (1.44)

2

2

22

a

bS =

a1=0 (1.45)

Of notable importance are the conditions under which these parameters are defined, namely, a1=0 or a2=0. These conditions are un-ambiguously realizable in the RF/microwave frequency bands, thereby making S parameters the most commonly used network parameters for microwave and RF work.

1.23.1.6 Inter-relationship among network parameters Of particular interest to the RF/microwave engineer are the formulas for converting S parameters into other parameters and vice-versa. For easy reference, these formulas are available in TBD.

1.23.2 Single-port networks

The Z,Y,h,ABCD, and S parameter definitions presented in the above section can be simplified for single-port networks, simply by omitting all references to the output port (port 2). For example, eqs. (1.42) to (1.43) would degenerate into a single equation

1111 aSb = (1.46) where S11 is still given by eq. (1.42), after ignoring reference to a2.


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Fig 1.1 A basic communication link

Fig 1.2 A two-port network

Transmitter (TX)

Propagation Medium

Receiver (RX)

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Eem Notes Chapter1 002