Antenna Basics Single Page

EMG4066 – ANTENNA AND PROPAGATION

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

1

Prerequisites for EMG4066 • Please drop this subject if you have not taken (or pass): – ECT1026 – Field Theory. – EMG2016 – Electromagnetic Theory. – PEM1026 - Engineering mathematics II. – PEM2036 - Engineering mathematics III. • Assumed prior knowledge: – Mathematics – Vector calculus, know what is gradient, divergence, curl, Stoke’s Theorem, Divergence Theorem. – Electrostatics – Coulomb’s Law, electric field, electric charge distribution, electric potential, Gauss’ Law. – Magnetostatic – Lorentz’s Force Law, Biot-Savart Law, magnetic vector potential, Ampere’s Law, Faraday’s Law. – Electrodynamics – Maxwell’s equations in time-domain and frequency-domain (time-harmonic forms), uniform plane waves propagation, refraction, reflection. Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

2

EMG4066 Main Topics Antenna Basics (Dr Fabian Kung) (1.5 - 2 weeks). Antenna Types (Dr Fabian Kung) (0.5-1 week). Antenna Arrays (Mr Gobi Vetharatnam) (1 week). Introduction to RADAR systems (Mr Gobi Vetharatnam) (1 week). • Radio wave propagation (Dr Deepak Kumar) (2 weeks).

• • • •

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

3

Main References/Textbooks • C. A. Balanis, “Antenna theory – analysis and design”, 3rd edition, 2005, John-Wiley & Sons. • W. Tomasi, “Electronic communication systems – fundamental through advanced”, 5th edition, 2003, Prentice Hall.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

4

EMG4066 Assessment • Lab 10% (AP1 – Antenna measurements, AP2 – Radar measurements at Applied Electromagnetic Lab, Level 3, FOE Building). • Assignment 15%. • Mid-term test 15% (30 Nov 2009, 7-8pm, covers antenna basics to antenna array). • Final 60% (covers all topics).

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

5

Antenna Basics

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

6

References • MAIN REFERENCES: • [1] D. K. Cheng, “Field and waves electromagnetic”, 2nd edition, 1989, Addison-Wesley. • [2] C. A. Balanis, “Antenna theory – analysis and design”, 3rd edition, 2005, John-Wiley & Sons. • OTHER REFERENCES: • [3] J. D. Kraus, “Antenna for all applications”, 2001, McGraw-Hill. • [4] S. Ramo, J. R. Whinnery, T. Van Duzer, “Field and waves in communication electronics”, 3rd edition, 1993, John-Wiley & Sons. • [5] Pass year EMG4066 notes. • [6] R. E. Collins, “Foundation for microwave engineering”, 2nd edition, 1992, McGraw-Hill.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

7

Agenda 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Definitions, usage and types of antenna. A quick review of Maxwell’s Equations. Mechanism of electromagnetic (EM) radiation. Finding the EM fields for antenna 1 - Potential theory for EM fields. Finding the EM fields for antenna 2 - Dynamic EM fields from elemental electric dipole and magnetic dipole. Finding the EM fields for antenna 3 – Radiation Integrals and Radiated EM fields for wire antenna. Antenna in transmit mode - Radiated power density, radiation intensity and antenna pattern. More antenna parameters – Directivity, beam solid angle, gain, antenna equivalent circuit. Antenna in receiving mode and Reciprocity Theorem – Antenna effective area. Friis transmission formula for transmit/receive system. Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

8

1. Definitions, Usage and Types of Antenna

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

9

What is an Antenna? • An antenna is... • That part of a transmitting system that is designed to generate propagating electromagnetic (EM) waves in free space. • The same structure can also be used to convert propagating electromagnetic waves in free space into voltage and current at the antenna terminals. • A transducer between a guided wave propagating in a transmission line and an electromagnetic wave propagating in an unbounded medium (usually free space), or vice versa. • Any conducting structure can launch/generate EM waves – or radiate – but when the “structure” is designed to radiate efficiently with directional and polarization properties suitable for the intended application – that “structure” is called an antenna. • An antenna can include purely metallic, purely dielectric or hybrid structures. Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

10

Usage of Antenna (1)

Horn antenna connected to rectangular waveguide

The figure shows how a wave is launched by a hornlike antenna, with the horn acting as a transition between the waveguide and free space.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

11

Usage of Antenna (2) • A pair of antenna used in a transmit/receive system, i.e. a wireless communication system. Electromagnetic

Antenna (EM) fields I1

I2

V1

V2 TX

RX

Near fields

Oct 2009

Far fields

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

12

Types of Radiating Structures (Antennas) NOTE: Usually contains TWO pieces of conductor

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

13

Types of Antenna (1) I1

Monople antenna

V1

Proprietary RF chipset

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

14

Types of Antenna (2) I1

Dipole antenna Folded dipole

V1

antenna

A variant of folded dipole on PCB (it’s actually a monople with two conductors) 2.4 GHz DSSS chipset

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

15

Types of Antenna (3) I1

Fractal patch

V1

antenna

Microstrip patch antenna

Coaxial to waveguide adapter

I1

Rectangular waveguide

Slot antenna

V1 Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

16

Types of Antenna (4) I1 V1

Horn antennas Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

17

Types of Antenna (5)

I1

Helix and fractal

V1

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

18

Types of Antenna (6) Microstrip line

Folded dipole antenna Microstrip patch antenna

Microstrip array and printed dipole array Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

19

Types of Antenna (7) Microstrip array Front Back

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

20

Types of Antenna (8) Ceramic chip antenna

Bluetooth chipset

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

21

Reasons for the Wide Variety of Antenna Types • • • • • • • •

Operating frequency. Power level (maximum voltage and current). Efficiency. Bandwidth or operating frequency range. Cost. Size. Gain. Technology.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

22

2. A Quick Review of Maxwell’s Equations

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

23

Review of Scalar and Vector Notations • • • • •

Suppose we have a variable, call it A. If A is a scalar, we would just write as A.r If A is a vector, we would write as A or A . If A is a unit vector (a vector with a magnitude of 1), we would write as Aˆ . In 3D Cartesian coordinate system, A and Aˆ would be defined as follows: Magnitude of A x-component magnitude z r ) ) ) 2+A 2+A 2 A = A A = A x + A y + A z x y z Az x y z

A vector A as depicted in Cartesian coordinate system

Ax x

A 0

Ay

Unit vector along x-axis ) ) ) Ax x + A y y + Az z Aˆ = A ) ) ) 1 = Ax x + Ay y + Az z Ax 2 + A y 2 + Az 2 y

(

)

Note: some books, like the textbook will use ax to denote unit vector in x-axis, and similar notation for unit vectors in y and z-axis.

r Another way to portray A: A = AAˆ = Aa A Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

24

Maxwell’s Equations (1) • How the physical quantities of electric charge q, electric current I, electric field E and magnetic field B (EM fields) behave and react in space are dictated by a set of natural laws called the Maxwell’s Equations. • Maxwell’s Equations can be expressed in differential or integral form. In differential form we use charge density ρv and current density J instead of q and I. In this chapter we will predominantly use Maxwell’s Equations in differential form. • In either differential or integral form, the parameters in Maxwell’s Equations are expressed as time-domain functions, since electric charge q, electric current I, E and B fields can change with time (e.g. they are function of time t) and location. • However in linear dielectric medium, when the parameters dependency on time is in sinusoidal form, the parameters can be expressed in timeharmonic form, where the electric charge, electric current, E and B fields are expressed as complex exponent in time or phasors. Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

25

Maxwell’s Equations (2) • Maxwell’s Equations tells us that the presence of electric charge causes electric field (E) to appear in the surrounding space. It also tells us that the presence of moving electric charge (current) produces magnetic field (B) in the surrounding space. • From the Lorentz Force Law, the electric field in turn produces a force on electric charge and the magnetic field in turn produces a force on moving electric charge. • Finally Maxwell’s Equations also describe the phenomenon of induction, in which time-varying electric field induces magnetic field, and time-varying magnetic field induces electric field. • These effects are summed up in 4 differential equations involving E, B, q and I (or ρv and J).

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

26

Maxwell Equations (Linear Medium) - TimeDomain Form (1) Faraday’s law

r

Each parameter depends on 4 independent variables

Where: E = E x (x, y, z , t )xˆ + E y (x, y, z , t ) yˆ + E z (x, y, z , t )zˆ

r r r ∂ ∇ × E = − B (2.1a) H = H x (x, y, z , t )xˆ + H y (x, y, z , t ) yˆ + H z (x, y, z , t )zˆ ∂t r r r ∂ r J = J x (x, y, z , t )xˆ + J y (x, y, z , t ) yˆ + J z (x, y, z , t )zˆ ∇ × H = J + D (2.1b) ρ v = ρ v ( x, y , z , t ) ∂t Unit vector in x-direction r Modified Ampere’s law In Cartesian coordinate system x component ∇ ⋅ D = ρv Gauss’s law (2.1c) r E – Electric field intensity No name, but can ∇⋅B = 0 H – Auxiliary magnetic field (2.1d) be called Gauss’s law for magnetic field

z

y x

Constitutive relations For linear medium Oct 2009

D – Electric flux B – Magnetic field intensity J – Current density ρv – Volume charge density εo – permittivity of free space (≅8.85412×10-12) µo – permeability of free space (4π×10-7) εr – relative permittivity µr – relative permeability

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

27

Maxwell Equations (Linear Medium) - TimeDomain Form (2) • Maxwell Equations as shown are actually a collection of 4 partial differential equations (PDE) that describe the physical relationship between electromagnetic (EM) fields, current and electric charge. • The Del operator is a shorthand for three-dimensional (3D) differentiation: ∇ = ∂∂x xˆ + ∂∂y yˆ + ∂∂z zˆ

)

(

• For instance consider the Faraday’s Law and Gauss’s Law, in Cartesian coordinate system: If you still remember Vector Calculus, the Curl is the limit of this operation:

Curl

xˆ r ∇× E = ∂

(

yˆ

∂x

∂ ∂y

Ex

Ey

zˆ

∂ =  ∂E z − ∂E y ∂z  ∂y ∂z

Ez

)

= − ∂ Bx x + B y yˆ + Bz zˆ Divergence ∂t r ∂E x ∂E y ∂E z

 r r  lim ∆S→ 0  1 ∫ E ⋅ dl   ∆S ∆C 

  ∂E x ∂E z   ∂E y ∂E x   xˆ +  −  yˆ +  ∂x − ∂y  zˆ ∂ z ∂ x     

Gradient

) ) ∇F = ∂F x + ∂F yˆ + ∂F z ∂x

∂y

∂z

 r r  ρ v If you still remember Vector Calculus, lim ∆V →0  1 ∫∫ E ⋅ ds  the Divergence is the limit of this operation:  ∆V ∆S  ∇⋅E = + + = ∂x ∂y ∂z ε Oct 2009 © Fabian Kung Wai Lee & Gobi Vetharatnam 2009 28

Maxwell Equations (Linear Medium) - TimeDomain Form (3) • The physical meaning of Maxwell’s Equations: Integral form

Faraday’s Law :

r r ∂ ∇× E = − B Differential form

∂t

Surface integration

r r ∂ E ⋅ d l = − ∫

r r ∫∫ B ⋅ ds

C

S

Line integration

∂t

B Induced E field (circular)

This means circular electric field E can be created by time-varying magnetic field (the magnetic field changes in value and direction) Modified Ampere’s Law:

r r r ∂ ∇× H = J +ε E Differential form

∂t

r r r ∂ H ⋅ d l = I + ε ∫

r r ∫∫ E ⋅ ds

C

S

∂t

Integral form

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

E

conductor

This means that circular magnetic field H can be created by both electric current (the current density) and time-varying electric field E (the electric field changes in value and direction) Oct 2009

I

Induced H field (circular) 29

Maxwell Equations (Linear Medium) - TimeDomain Form (4) Gauss’s Law :

r ρv ∇⋅E = ε

Differential form

r r 1 Q ∫∫ E ⋅ ds = ε ∫∫∫ ρ v dV = ε S

E

V

Integral form

This means electric field E can also be created from electric charge, and the field pattern is in a radial direction

Electric charge

Gauss’s Law for magnetic field:

r ∇⋅B = 0 Differential form

Oct 2009

r r ∫∫ B ⋅ ds = 0

Make sure you ‘grasp’ the insights of the concepts of S Integral form field properly. An important concept is ‘action at a distant’. This means for the time being there is no natural magnetic charges to create magnetic field in radial direction!!! © Fabian Kung Wai Lee & Gobi Vetharatnam 2009

30

Review of Phasor (1) • In engineering we usually deal with signals that changes with time in a sinusoidal manner. This is because many non-sinusoidal signals can be expressed in terms of sinusoidal components by the use of Fourier v1(t) Series and Fourier Transform. Vo Frequency • For example a voltage: t v1 (t ) = Vo cos(ωt + θ ) θ

Phase

Magnitude

2π

ω

• This can be expressed as complex exponent via Euler’s formula: e jα = cos α + j sin α

We normally use small letter and italic font to represent time-domain variable

Phasor

{

} {

v1 (t ) = Re Vo e j (ωt +θ ) = Re Vo e jθ e jωt

}

• The term V1 = Vo e jθ is called the phasor, or time-harmonic form. • Similarly, given a phasor, we can obtain the time-domain form as follows: – Multiply the phasor with e jωt . – Take the real part of the product. Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

31

Review of Phasor (2) • Why use phasor ? • In many engineering problems we are only interested in the steadystate sinusoidal response of a linear system, which can be conveniently represented in phasor form. • Moreover using the phasor notation simplifies the integral-differential equations describing a physical system. • In particular the differentiation and integration with respect to time t becomes multiplication and division with jω respectively in phasor. Time-domain quantity

∂v → jωV ∂t

Phasor/time-harmonic quantity

1 V → vdt ∫ jω

• For more discussion on the theory of phasor analysis and Fourier Transform, consult your pass year notes and textbooks on circuit and signal analysis.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

32

Review of Phasor (3) • Vector quantity such as E and H fields, J can also be expressed as phasors provided all their components’ time dependency are sinusoidal. • For example for a sinusoidal E field:

r E(x, y, z, t ) =

) Ex (x, y, z)cos(ωt +θ )xˆ + Ey (x, y, z)cos(ωt +θ ) y + Ez (x, y, z)cos(ωt +θ )zˆ

{

}

• The Re E y e jθ e jωt r phasor is given by: E ( x, y , z ) = E (x, y , z )e jθ xˆ + E (x, y, z )e jθ y + E (x, y, z )e jθ zˆ x

y

z

) ) ) = Ex x + E y y + Ez z Normally straight and capital letter is used to represent phasor

• Notice that the phasor now depends on (x,y,z) only (it can also depends on frequency ω), and it is also a vector. Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

33

Maxwell Equations (Linear Medium) - TimeHarmonic Form (1) • For sinusoidal variations with time t, we substitute the phasors for E, H, J and ρ into Maxwell’s Equations, the result are Maxwell’s Equations in Each parameter depends on 3 independent variables time-harmonic form. Where: r ∂ → jω E = E x (x, y, z )xˆ + E y (x, y, z ) yˆ + E z (x, y, z )zˆ ∂t

Faraday’s Law Modified Ampere’s Law Gauss’s Law Gauss’s Law for magnetic field

r r ∇ × E = − jωB (2.2a) r r r ∇ × H = J + jωD (2.2b) r ∇ ⋅ D = ρ v (2.2c) r ∇⋅B = 0 (2.2d)

r H = H x (x, y, z )xˆ + H y (x, y, z ) yˆ + H z (x, y, z )zˆ r J = J x (x, y, z )xˆ + J y (x, y, z ) yˆ + J z (x, y, z )zˆ

ρ v = ρ v (x, y, z ) E – Electric field intensity

Constitutive relations For linear medium Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

H – Auxiliary magnetic field D – Electric flux B – Magnetic field intensity J – Current density ρv- Volume charge density εo – permittivity of free space (≅8.85412×10-12) µo – permeability of free space (4π×10-7) εr – relative permittivity µr – relative permeability 34

3. Mechanism of Electromagnetic (EM) Radiation

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

35

Self-Sustaining EM Fields • Maxwell’s Equations show that time-varying E field can induce solenoidal H field. • Similarly time-varying H field will induce solenoidal E field. • It is this symmetry nature of time-varying EM fields which result in selfsustaining EM field in free space. When the variation with respect to time is sinusoidal, this self-sustaining EM field is usually called a propagating EM wave or radiowave.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

36

Example 3.1 – Radiation from Dipole Antenna

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

37

Example 3.2 – Radiation from Horn Antenna Driven by Rectangular Waveguide • To be shown during lecture (hand-drawn).

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

38

4. Finding the EM fields for Antenna 1 - Potential Theory for Electromagnetic fields

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

39

Introduction • In the last section we only explain the possibility of generating a selfsustaining EM fields from time-varying currents in an antenna. • In this section systematic procedures for finding the approximate expressions for the electric and magnetic fields from an antenna is presented. • From our knowledge of Maxwell’s Equations, it is obvious that to find the EM fields, we need to know how the current and charge are distributed and change with time in an antenna. • However to find the EM fields directly via Maxwell’s Equations in it’s original form is difficult, since E and H fields being vectors, has three components each, that brings the total unknowns to six. • A more systematic approach to finding the E and H fields is to write the fields in terms of a auxiliary (or secondary) quantity, traditionally called potentials. Express the potentials in terms of charge and current, and obtain the EM fields from the potentials. Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

40

Maxwell’s Equations Under Static Condition (1) • In static case the EM fields, J and ρv are constant with respect to time t, they are no longer function of time, i.e. frequency is zero, ω = 0. r r E ( x , y , z , t ) → E ( x, y , z )

• Maxwell’s Equations under static condition degenerate to the following form: r

r r ∇ × E = − j ωB r r r ∇ × H = J + jωD r ∇ ⋅ D = ρv r ∇⋅B = 0

For linear medium with ω → 0

∇×E = 0 r r ∇×H = J r ∇ ⋅ D = ρv r ∇⋅B = 0

We observe that: • There is a symmetry between the Electrostatics and Magnetostatics equations. •The E and H fields are no longer interconnected (decoupled). Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

Electrostatics

r ∇ × E = 0 (4.1a) r ∇ ⋅ εE = ρ v (4.1b)

( )

Magnetostatics

r r ∇ × H = J (4.2a) r ∇ ⋅ µH = 0 (4.2b)

( )

41

Electrostatic, Charge and Scalar Potential (1) • We have learnt in field theory and basic EM theory that under electrostatic condition, an auxiliary quantity called Electric Potential (V) can be defined, and it is related to electric field E. Electric potential (Energy needed to bring 1 Coulomb of positive charge from a reference point to the observation point under electric field)

Electrostatics

r ∇×E = 0 r ∇ ⋅ εE = ρ v

( )

r E = −∇V

Poisson’s Equation for Electrostatic (under linear homogeneous medium)

(4.3)

2

∇ V=−

r r V = −∫ E ⋅ d l

Electrostatics in terms of V

(4.4)

V(R ) = Potential at observation point, reference at infinity

Oct 2009

ρv εo

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

1 4πε o

∫∫∫

ρv r Volume D

dv

'

Solution to Poisson’s Equation 42

Electrostatic, Charge and Scalar Potential (2) • A diagram should clear up any confusion with regards to the notations of finding electric potential V from a given charge distribution. Integration z domain for dv’

Observation point P(x,y,z) or P(R,θ,φ) R

V(R ) =

r

1 4πε o

∫∫∫

ρv r Volume D

dv

'

ρv(r’) r’

x

A distribution of electric charge, ρv

Volume D y

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

43

Magnetostatic, Current and Vector Potential (1) • In a similar manner we have also learnt that static magnetic field can also be expressed in terms of another auxiliary quantity, called magnetic vector potential A. Magnetic vector

Magnetostatics

r ∇ ⋅ µH = 0 r r ∇×H = J

r r potential H = µ1 ∇ × A (4.5) r r r r ∫ A ⋅ d l = ∫∫ µH ⋅ d s Closed surface S

( )

Vector Poisson Equation for A

r r ∇ A = −µ o J 2

(4.6)

r A(R ) =

µo 4π

r J r Volume

∫∫∫

dv '

Solution to vector Poisson’s Equation Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

44

Magnetostatic, Current and Vector Potential (2) • Similar to Electrostatic we can find the vector potential A from a given current distribution. Integration z domain for dv’

Observation point P(x,y,z) or P(R,θ,φ) R

r A(R ) =

r

µo 4π

r J r Volume D

∫∫∫

dv

'

J(r’) r’

x

A distribution of electric current, J

Volume D y

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

45

Summary- Potentials in Electrostatics and Magnetostatics • From electrostatics and magnetostatics we have seen that E and H fields can be expressed in terms of potentials, called scalar potential V and vector potential A respectively.

r E = −∇V r r B = ∇×A

(4.3) or

r r H = µ1 ∇ × A

(4.5)

• Here we observe that both E and H fields are independent of each other, E depends on V, while H depends only on A. • The electric potential V depends on the charge distribution of the system, while the vector potential A depends on the current distribution of the system. Under static condition, ρv and J are independent and thus V and A are independent. • Knowing the charge and current in a system allows us to work out the E and H field of static system. Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

46

Potentials in Electrodynamics (1) • We shall now see that under electrodynamics E and H will depend on both scalar and vector potentials. r • Under electrodynamics E ≠ −∇V because curl of E is no longer zero. r r • However divergence of B remains as zero, thus B = ∇ × A is still valid. • By putting (4.5) into Faraday’s Law of Maxwell’s Equations:

r r ∇ × E = − jω ∇ × A r r ⇒ ∇ × E + jωA = 0

(

(

)

)

(4.6)

• And using the vector identity for zero curl, the above can be expressed as gradient of a scalar function V. To be consistent with electrostatic, we also include the negative multiplier:

r r E + jωA = −∇V

Oct 2009

(4.7)

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

Here I want to reiterate that we are using time-harmonic form of Maxwell’s Equations

47

Potentials in Electrodynamics (2) • Therefore under electrodynamics E and H can be expressed as:

r r E = −∇V − jωA r r H = µ1 ∇ × A

(4.8a) (4.8b)

Potentials formulation in electrodynamics

• As usual V and A are the scalar and vector potentials respectively, but these now apply to a system with electric charge and electric current that varies with time, e.g. in a sinusoidal fashion.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

48

Wave Equations for Potentials (1) • Also recall that under electrostatics and magnetics both V and A fulfill the scalar and vector versions of the Poisson Equations, repeated again: ρv 2

∇ V=−

ε

r r ∇ A = −µJ 2

(4.4)

(4.6)

• This can be extended to electrodynamics condition. For instance putting (4.8a) for E into Gauss’s Law of Maxwell’s Equations:

r r ∇ ⋅ εE = ε∇ ⋅ − ∇V − jωA = ρ v r 2 ⇒ ∇ V + jω ∇ ⋅ A = − 1ε ρ v

(

( )

(

)

)

(4.9a)

This is a 2nd order linear partial differential equation. Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

49

Wave Equations for Potentials (2) • In a similar manner using the modified Ampere’s Law in Maxwell’s Equations, a ‘constraint’ for A in the form of 2nd order PDE can be obtained:

r r r r 2 ∇ A + µεω A − ∇ ∇ ⋅ A + jµεωV = −µJ

(

2

) (

)

(4.9b)

• As usual we observed that (4.9a) and (4.9b) are coupled PDEs, containing both A and V terms. We would like to decouple them. • At this juncture we should pause and ponder the reason we use potential formulation. Note that we manage to reduce the complexity of the problem slightly, instead of 6 parameters (3 each for E and H fields in 3D space), we manage to reduce the unknown parameters to 4 (3 for A and 1 for V in 3D space).

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

50

Wave Equations for Potentials (3) • Now take note that equations (4.8a) and (4.8b) do not uniquely define V and A. We are free to impose extra conditions on V and A as long as nothing happens to E and H. • The extra conditions on V and A are called Gauge Transformation (see reference books for more info). • Here I just wish draw your attention to a theory on vector field, the Hemholtz Theorem which states that to uniquely define a vector field (A in this case), one needs to define both it’s divergence and curl. • Here we have already define the curl of A in (4.8b). Now let’s impose r further condition on A, in the form of:

∇ ⋅ A = − jωµεV

(4.10)

This is called the Lorentz’s Gauge, note that many type of gauge can be defined. r For example ∇ ⋅ A = 0 is called the Coulomb’s Gauge. Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

51

Wave Equations for Potentials (4) • With the Lorentz’s Gauge, equations (4.9a) and (4.9b) can be decoupled and we finally obtained the following PDEs for the potentials under electrodynamics condition: 2

2

∇ V + k V = − ε1 ρ v r r r 2 2 ∇ A + k A = − µJ r ∇ ⋅ A = − jωµεV and

k = ω µε

(4.11a) (4.11b) (4.11c) (4.11d)

• The above is what we called the Wave Equations for Potentials in Electrodynamics. • Note with the Lorentz Gauge, E and H fields can be specified entirely in r r r vector potential A. jω

(

)

E = − k 2 ∇ ∇ ⋅ A − jωA

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

52

Solutions for Wave Equations for Potentials (1) • Now we would like to know the solutions for wave equations (4.11a) and (4.11b). • Here we do not have the space to outline the procedures of finding the expressions for V and A that fulfill equations (4.11). • The motivated reader can refer to the standard electromagnetic books or to the excellence texts by: – [4], Chapter 12. – C.A. Balanis, “Advanced engineering electromagnetics”, 1989 John Wiley or [2]. – [1], Chapter 7 (this is presented in time-domain). • In summary, the solution to (4.11) for V and A, expressed in terms of charge and current density can be obtained using “Green Function Theorem” for Partial Differential Equations (PDE).

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

53

Solutions for Wave Equations for Potentials (2) • The solutions to (4.11) for dynamic case are quite similar in form to the Electrostatic and Magnetostatic case: Note that under static

V ( x, y , z ) =

∫∫∫

1 4πε

r A ( x, y , z ) =

µ 4π

ρ v (r ') r

∫∫∫

r J (r ') r

e

− jkr

e

dv'

− jkr

dv'

(4.12a)

condition, (1.9a) and (1.9b) reduced to the solutions for electrostatics and magnetostatics.

(4.12b)

• Contrast (4.12a) and (4.12b) with the expressions for V and A for electrostatics and magnetostatics. • In time-domain (4.12a) and (4.12b) can be expressed as:

V ( x, y , z , t ) =

r A( x, y, z , t ) =

1 4πε

µ 4π

ρ v (r ',t − cr )

∫∫∫ r

∫∫∫

r

J (r ',t − cr ) r

dv'

(4.13a)

dv'

(4.13b)

These are called the Retarded Potentials

• Where c is the speed of light in the medium. Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

54

Solutions for Wave Equations for Potentials (3) • As usual, a diagram should clear up any confusion with regards to the notations. This is shown for V, and should apply for A too. Integration z domain for dv’

Observation point P(x,y,z) R

r

V ( x, y , z ) =

1 4πε

∫∫∫

ρ v (r ' ) r

e − jkr dv'

ρv(r’) r’

A distribution of electric charge, ρv

V ( x, y , z , t ) =

x y

Oct 2009

or

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

1 4πε

∫∫∫

ρ v (r ',t − cr ) r

55

dv'

Main Ideas • What equations (4.12a) and (4.12b) say is that if we know the charge and current distribution in a system, we can predict the potentials and subsequently the electromagnetic fields at any observation point. • When there is a change in the charge or current distribution, this will affect the potentials/fields at the observation point. However the affect will not be immediate, but delayed by a factor of r/c in time-domain or kr in frequency domain for each point source. • We can explain this physically by noting that a ‘signal’ cannot travel faster than the speed-of-light. • Thus in general, if we know the charge and current in a structure, we can predict the EM fields with time retardation taken into account, and this is the basis of finding the radiating fields from an antenna.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

56

5. Finding the EM fields for Antenna 2 - Dynamic EM Fields from Elemental Electric Dipole and Magnetic Dipole

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

57

Short Electric Dipole • Consider a Short Electric Dipole (since this structure produce mainly electric field in it’s vicinity) with oscillating current flowing along it’s axis. Electric charges exist at both ends to fulfill the charge continuity principle. • The short, or elemental electric dipole is also called a Hertzian dipole, and it is a form of antenna. +

∆l -

Point charges at both ends, ±q(t).

Q∆l = Electric dipole moment

Electric current flows uniformly along conductor, which is very thin, i(t).

i (t ) = I o cos(ωt )

or

i (t ) = ±

or

dq dt

Time-domain Oct 2009

“Short” here means ∆l < 0.1λ

I = I oe j 0 Io Q=± jω

(5.1a) (5.1b)

Frequency-domain © Fabian Kung Wai Lee & Gobi Vetharatnam 2009

58

Potentials for Elemental Electric Dipole • Consider an elemental electric dipole oriented as shown. From (4.12), the potentials at the observation point is then given by: ∆l = ∆z

z

P(observation point)

R r1

r r2

r A(R ) =

µI o ∆z e − jkr 4π r

V (R ) =

Q 4πε

( )zˆ = A zˆ (5.2a) ( )− ( ) (5.2b) e − jkr1 r1

z

Q 4πε

e − jkr2 r2

NOTE: r’

The parameters r1 and r2 can be expressed Elemental electric dipole in terms of r, ∆z. We will not do that now

x y

Oct 2009

as potential V can be written in terms of A (4.10), so it is not needed. We will see very soon that V is not needed at all as the elemental electric dipole is normally not used alone.

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

59

EM Fields from Elemental Electric Dipole • For simplicity assume that r’=0, thus R = r and the dipole is at the center of the origin. • The vector potential A can be written in spherical coordinate system as: z

r A(r , θ , φ ) = Ar rˆ + Aθ θˆ + Aφφˆ

= ( Az cos θ )rˆ + (− Az sin θ )θˆ • Using the relation between A and H, at point P:

r r H(r ) = µ1 ∇ × A

= − I4o π∆z k 2 sin θ

(5.3a)

[

1 jkr

r

]

+ ( jkr1 )2 e − jkrφˆ

• Knowing H we can find E from Maxwell’s Equations.

r E(r ) =

1 jωε

r ∇×H

= − I4o π∆z ηk 2 2 cos θ Oct 2009

x

[

1 ( jkr )2

P

θ

]

+ ( jkr1 )3 e − jkr rˆ − I4o π∆z ηk 2 sin θ

[

1 jkr

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

φ y

]

+ ( jkr1 )2 + ( jkr1 )3 e − jkrθˆ (5.3b) 60

Example 5.1 • An example of EM fields from a elemental electric dipole at two difference instance in time. • Equations (5.3a) and (5.3b) for H and E fields can be converted into time-domain respectively and plot at t = 0 and t = 0.25T, whre T =1/f. E fields at t=0.25T

E fields at t=0 Instantaneous E field

Vertical Plane

Oct 2009

Source: S. A. Schelkunoff and H. T. Friis, “Antenna: theory and practice”, 1952 John-Wiley © Fabian Kung Wai Lee & Gobi Vetharatnam 2009 and Sons.

61

Near and Far Fields (1) • The EM fields can be classified depending on the distance of the observation point to the center of the elemental electric dipole. • Typically reactive near field is defined as the EM fields at observation 2πr point where: (5.4a)

kr =

λ

<< 1

• Far field is defined as the EM fields at observation point where:

kr =

2πr

λ

>> 1

(5.4b)

• In between reactive near field and far field is the radiating near field.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

62

Near and Far Fields (2)

Radiating near field (Fresnel) region (kr > 1)

Rnear

Antenna Rfar

No abrupt changes in the field configurations are noted as the boundaries are crossed – but there are distinct differences between the fields

Reactive near field region (kr << 1) Far field (Fraunhofer) region (kr >> 1) Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

63

Near and Far Fields (3) • For reactive near field:

r H (r ) ≅ − I4o π∆z k 2 sin θ

[ ]φˆ

r E (r ) ≅ − I4o π∆z ηk 2 2 cos θ

[ ]rˆ −

(5.5a)

1 ( jkr )2

1 ( jkr )3

I o ∆z 4π

ηk 2 sin θ

[ ]θˆ 1 ( jkr )3

• For far field:

r H (r ) ≅ − I4o π∆z k 2 sin θ r E (r ) ≅ − I4o π∆z ηk 2 sin θ

Oct 2009

[ ]e 1 jkr

[ ]e 1 jkr

− jkr

θˆ

− jkr

φˆ

(5.6a) (5.6b)

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

(5.5b)

NOTE: Far E and H fields perpendicular to each other and in phase, this indicates the Poynting Vector is non-zero. Also

r E (r )far -field =η r H (r )far -field

64

Why Consider Elemental Electric Dipole? • In general any conductor can be considered as composing of many elemental electric dipole connected together. • Thus the EM field generated by the long conductor can be obtained by superposition of the fields from the individual elemental electric dipole. Long thin conductor

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

65

Elemental Magnetic Dipole • A current loop is called a Magnetic Dipole (because it produce magnetic field mainly). If the loop area is very small, it is called elemental magnetic dipole. • When the current within the magnetic dipole is time-varying, the resulting H field in it’s vicinity is also time-varying and this induces a z propagating EM wave.

Current loop with area a

θ Io

x

Observation point P r

φ y

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

66

EM Fields for Elemental Magnetic Dipole • Using similar procedures, e.g. finding the A of the current loop, and then using expressions (4.8a) and (4.8b) with reasonable approximation, one can show that the far-field E and H for the elemental magnetic dipole are given by [1]:

r H (r ) ≅ − ωµ4πηo m sin θ [ kr ]e − jkrθˆ r E (r ) ≅ ωµ4πo m k sin θ [1r ]e − jkrφˆ • As in the electric dipole,

Area

(5.7a) (5.7b)

r E (r )far -field =η r H (r )far -field

m = Ioa Magnetic dipole moment

• Examination of (5.6) and (5.7) reveals that electric and magnetic dipole have the same pattern function and are in both space and time quadrature. This means it is possible to combine electric and magnetic dipoles to form an antenna that produces circular polarization EM wave. Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

67

Concluding Remarks • Every antenna structure can be decomposed into many short electric and magnetic dipoles. • Thus understanding the EM field of these elemental radiating structures is important. • Due to time constraint, we shall not discuss much on elemental magnetic dipole. We shall only consider in detail antenna whose structure can be decomposed into many elemental electric dipole.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

68

6. Finding the EM fields for Antenna 3 - Radiation Integrals and Radiated EM Fields for Wire Antenna

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

69

EM Fields from General Wire Structures (1) • Consider a long conductor oriented along z direction. • By segmenting it into short segments of ∆l, the total EM fields can be written using superposition principle and (5.3a), (5.3b): z

Long thin conductor

segment 1

θ1

segment 2

r2

segment 3

∆z

r1

P

r3

N r k2  H (P ) ≅ − 4π ∑ I oi sin θ i  i =1

1 jkri

]

 + ( jkr1 )2 e − jkri ∆z φˆ i 

[

]

N r  − jkri ηk 2  1 1 E (P ) ≅ − 2π ∑ I oi cos θ i ( jkr )2 + ( jkr )3 e ∆z rˆ i i  i =1  N 2  ˆ − jkri ηk 1 1 1 − 4π ∑ I oi sin θ i jkri + ( jkr )2 + ( jkr )3 e ∆z θ i i  i =1 

[

segment N

[

]

0 Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

70

EM Fields from General Wire Structures (2) • In the limit when N→∞ or ∆z→0, the summation operation becomes integration: We need to know this!

r k2 H ( P ) = − 4π

{∫ I (z )sin(θ (z ))[

r ηk 2 E ( P ) ≅ − 2π

{∫ I (z )cos(θ (z ))[

−

ηk 2 4π

L

o

0

L

0

0

o

1 ( jkr )2

o

{∫ I (z )sin(θ (z ))[ L

1 jkr

1 jkr

]

}

(6.1a)

]

}

(6.1b)

+ ( jkr1 )2 e − jkr dz φˆ + ( jkr1 )3 e − jkr dz rˆ

]

}

+ ( jkr1 )2 + ( jkr1 )3 e − jkr dz θˆ

• Of course to be able to use equations (6.1a) and (6.1b) effectively to predict the EM fields created by a thin, long conductor, we need information of the electric current distribution along the conductor.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

71

Far Field for General Wire Structures • In general we are more interested in the far EM field for a structure, as the far field represents the radiation field.

r k2 H ( P ) = − 4π r ηk 2 E ( P ) ≅ − 4π

{∫ I (z )sin(θ (z )) L

0

o

{∫ I (z )sin(θ (z )) L

0

o

1 jkr

1 jkr

}

e − jkr dz φˆ

}

e − jkr dz θˆ

(6.2a) Radiation Integrals

(6.2b)

• The above integral, representing far-field E and H in terms of the current in the conductor are called the Radiation Integrals. • An important observation above is that the ratio of |E| over |H| for far field is a fixed constant, called the Wave Impedance:

r E (P ) =η r H (P ) Oct 2009

(6.3)

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

72

Example 6.1 – Far Field for Dipole Antenna • We would like to find the far field for a dipole structure as shown below. • The determination of the exact current distribution on the dipole conductors subjected to Maxwell’s Equation is a very difficult boundary value problem even if the wire is assumed to be perfectly conducting. • Usually numerical method like Method of Moments (MoM) and Finite Element Method (FEM) is employed to find the exact current distribution (see [2]). z Dipole h

i(t) x h

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

73

Example 6.1 Cont… • For our purposes the knowledge of the exact current distribution on the linear antenna is not of prime importance. A good estimate will give us considerable useful information on the radiation characteristics of the antenna. • Here we assume a sinusoidal current distribution on a very thin, straight dipole. Such a current distribution constitutes a type of standing wave over the dipole and is a good approximation. z

Current phasor

I (z ) = I m sin k (h − z )  I m sin k (h − z ) z > 0 =   I m sin k (h + z ) z < 0

Common sense tells us that current magnitude must be large here and approach zero at the tips of the conductor

i(t) x

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

74

Example 6.1 Cont… • Using (6.2b):

z

r ηk 2 E ( P ) ≅ − 4π

−h

}

1 I ( z )sin (θ ( z )) jkr e − jkr dz θˆ

 h I sin (k (h − z ))sin (θ ( z )) 1 e − jkr dz  m jkr ˆ ηk 2  ∫0 = − 4π  0 θ 1 + ∫ I m sin (k (h + z ))sin (θ (z )) jkr e − jkr dz   −h  θ

i(t)

{∫

h

P (observation point)

r R

z x

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

75

Example 6.1 Cont… • The integration is still difficult to perform, we can make further assumption by noting that since R >> h, the angle θ can be considered constant along the dipole. • Furthermore r and R can be assumed to be parallel. z

r ≅ R − z cos θ θ

r R

i(t) z

x

r ηk 2 I m sin θ E ( P ) ≅ − 4π

Oct 2009

This term can be ignored without incurring much error − jk ( R − z cosθ )  h sin (k (h − z ))  1 e dz jk ( R − z cosθ ) ∫0 ˆ  0 θ + ∫ sin (k (h + z )) jk ( R −1z cosθ ) e − jk ( R − z cosθ )dz   −h 

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

76

Example 6.1 Cont… • Therefore

 h sin (k (h − z )) 1 e − jk ( R − z cosθ )dz  r jkR ˆ ηk 2 I m sin θ  ∫0 E ( P ) ≅ − 4π  0 θ 1 + ∫ sin (k (h + z )) jkR e − jk ( R − z cosθ )dz   −h   h sin (k (h − z ))e jk ( z cosθ )dz  ˆ jηkI m sin θ − jkR  ∫0 = 4πR e  0 θ + ∫ sin (k (h + z ))e jk ( z cosθ )dz   −h  jηkI m sin θ − jkR h = e sin (k (h − z )) cos(kz cos θ )dz θˆ

{∫

2πR

r E (P ) ≅

[

}

0

jηkI m 2πR

e

− jkR

]{

}θˆ

(6.4a)

[

e − jkR

]{

}φˆ

(6.4b)

cos ( kh cosθ )− cos kh sin θ

• Using (6.3):

r H (P ) ≅

Oct 2009

jkI m 2πR

cos ( kh cosθ )− cos kh sin θ

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

77

Example 6.1 Cont… • For the special case of half-wavelength dipole, 2h = λ/2, the far fields become: r jηkI m − jkR cos ( π2 cosθ ) ˆ (6.5a) E 1 λ P ≅ 2πR e sin θ 2

( )

[

r H 1 λ (P ) ≅

[

2

Oct 2009

]{

jkI m 2πR

e

− jkR

]{

cos ( π2 cosθ sin θ

}θ r ) ˆ E (P ) ˆ }φ = η φ

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

(6.5b)

78

7. Antenna in Transmit Mode - Radiated Power Density, Radiation Intensity and Radiation/Antenna Pattern

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

79

Radiated Power Density (S) • For application involving antenna, we are only interested in the EM fields at the far-field condition. • We can see that the E and H fields are perpendicular to each other under far-field condition, and their vector product produces a positive power flow from the antenna, according to Poynting Theorem. • The time-average power density (e.g. power per unit area) of the EM fields in a direction n (where n is a unit vector) is given by:

r r S = 12 Re E × H * ⋅ nˆ

{

}

(7.1) Denotes complex conjugate

• For antenna, we are usually interested in S at various direction, with nˆ equals to the radial direction in spherical coordinate system. • When equation (7.1) is applied to far field of a radiating structure with nˆ = rˆ , the resulting S is called the Radiated Power Density. Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

80

Radiated Power Density for Hertzian Dipole • For the Hertzian Dipole, the radiated power density is given by:

r r S = 12 Re E × H * ⋅ nˆ

{

= = • Or

I o 2 ∆z 2 32π 2

ηk

}

4

( )Re[(e (sin θ )2

− jkr − j π2

e

k 2r 2

I o 2 ∆z 2ηk 2 (sin θ )2 32π 2

S (θ , φ , r ) =

( ) W/m 1 r2

2

I o 2 ∆z 2ηk 2 (sin θ )2 32π

)]

)(

jkr j π2 ˆ ˆ θ × e e φ ⋅ rˆ

2

( ) W/m 1 r2

2

• This can be written in the form:

S (θ , φ , r ) = S o (r )F (θ , φ ) W/m 2 where S o (r ) = Oct 2009

(7.3a)

The pattern function I o2 ∆z 2ηk 2 32π 2 r 2

F (θ , φ ) = (sin θ )

2

(7.2) F(θ,φ) represents the normalized radiated power density at certain distance from the antenna.

(7.3b)

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

81

Radiation Pattern • We have seen in the previous slide the radiated power density depends on direction. • The normalized three-dimensional plot showing the relative strength of the radiation (i.e. it’s radiation power density) at different observation direction with distance fixed is given by F(θ,φ). • The function F(θ,φ) is called the Radiation Pattern or Pattern Function of an antenna. • The radiation pattern can be shown in two-dimension at certain ‘cutplane’.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

82

Example 7.1 – Radiation Pattern of Short Electric Dipole S (θ , φ , r ) = S o (r )F (θ , φ ) W/m 2

z

where

θ

S o (r ) =

I o2 ∆z 2ηk 2 2 2

32π r

F (θ , φ ) = (sin θ )

2

When φ is constant (E-plane pattern)

r When θ is π/2 (H-plane pattern)

φ

x

y

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

83

E-plane and H-plane • Is defined only for single linear polarized antenna. • The radiation pattern that contains the electric field is called the E-plane cut or E-plane pattern. • Automatically the other plane, which contains the Magnetic field is called the H-plane. For the short/Hertzian dipole, the left pattern cut is the Eplane and the right is H-plane

E- and H-plane does not mean anything for a dual polarized or circularly polarized antenna!

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

84

Solid Angle • A Solid Angle is like an angle in 3 dimension. It is defined as shown below, with a unit of Steradian. A r

Solid Angle Ω =

A r2

Steradian

(4.4)

• A sphere has a solid angle of 4π Steradian.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

85

Radiation Intensity (U) (1) • In describing the radiation property of the antenna another parameter is often used to complement the radiated power density S. • We know that S represents the time-average power per unit area along the observation direction. It is a scalar quantity. • A parameter, known as Radiation Intensity (U) can be used in place of S, it is equivalent to time-average power per unit solid angle along the radial direction only.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

86

Radiation Intensity (2) • The radiation intensity U is formally defined as:

U (θ , φ ) = =

Power Area r2

Radiated Power along observation direction Unit solid angle along observation direction

2 2 ) = ( Power r = r S (θ , φ ) Area

(7.5)

• U (θ,φ) has the benefit of being a scalar quantity AND independent of observation distance, provided the EM field is in Far-Field mode. • U (θ,φ) is usually used in place of F(θ,φ) to describe the radiation pattern of an antenna. • Here we will use either one, distinguished by their respective symbols.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

87

Total Radiated Power • The antenna can be considered as at the center of the coordinate, being enclosed by a sphere. • The total radiated power (Prad) from an antenna can be computed by integrating the radiated power density or radiated intensity over all direction on the sphere. Sphere

∫∫ S (r ,θ , φ ) ⋅ dA = ∫∫π U (θ , φ )⋅ dΩ

Prad =

A sphere enclosing the antenna

=∫

2π

0

π

∫ U (θ , φ )sin θdθdφ

Oct 2009

0

r

4

(7.6)

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

dA

Antenna

88

Example 7.2 – Total Radiated Power by Hertzian Dipole • The radiation intensity U of Hertzian Dipole is:

U (θ , φ ) = • Thus

Prad = ∫

2π

0

=∫

2π

0

I o 2 ∆z 2ηk 2 (sin θ )2 32π 2 π

∫ U (θ , φ )sin θdθdφ 0

π I 2 ∆z 2ηk 2 o 32π 2 0

∫

(sin θ )3 dθdφ

I o2 2 = 80π 2 ( ∆λz ) 2

[

Oct 2009

W/Steradian

]

(7.7)

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

89

Exercise 7.1 • (a) Find the radiated power density and radiation intensity of a halfwave dipole. • (b) Sketch it’s radiation pattern. • (c) Also find it’s total radiated power. Radiated power density and intensity of half-wave dipole, using (6.5a) and (6.5b):

S dipole =

1 2

r r* E × H ⋅ rˆ ≅

(

⇒ S dipole =

)

{

[

jηkI m 2πR

}

2 ηk 2 I m 2 cos ( π2 cosθ ) sin θ 4π 2 R 2

So(R)

e

− jkR

][

− jkI m 2πR

U dipole =

e

jkR

]{

} (θˆ × φˆ ⋅ rˆ) ) }

cos ( π2 cosθ ) 2 sin θ

{

ηk 2 I m 2 cos ( π2 cosθ sin θ 4π 2

2

F(θ,φ)

ηk 2 I m 2 2π π cos (π cosθ ) 2 sin θdθdφ Total radiated power Prad = ∫0 ∫0 U (θ , φ )sin θdθdφ = 4π 2 ∫0 ∫0 sinθ of half-wave dipole: π cos (π cosθ ) η k 2 I m 2 2π ηk 2 I m 2 π cos (π cosθ ) ⇒ Prad = dφ ∫ dθ = dθ sin θ sin θ 2 ∫ ∫ 0 0 0 4π 2π ηk 2 I m 2 ⇒ Prad = 1.219 Evaluated using numerical 2π integration Oct 2009 © Fabian Kung Wai Lee & Gobi Vetharatnam 2009 2π

(

π

2

)

2

2

2

2

90

Exercise 7.1 Cont… • The radiation pattern of half-wave dipole.

When φ is constant (E-plane pattern) The dotted blue line is the E-plane pattern for Hertzian Dipole, for comparison. Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

When θ is π/2 (H-plane pattern)

91

Example 7.3 – Radiation Pattern for a Horn Antenna

The main lobe

3D pattern

Side lobes

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

This is the spatial distribution of power radiated – also called power pattern

92

Major Lobe and Minor Lobe • Major lobe is also called main lobe. • Defined as “the radiation lobe containing the direction of maximum radiation”. • In certain antennas, such as multi-lobed or split beam antennas, there may exist more than one major lobe. • Minor lobe - A radiation lobe in any direction other than that of the major lobe. • When its adjacent to the main lobe its called side lobe. • Side lobe level – maximum relative directivity of the highest side lobe with respect to the maximum directivity of the antenna. • Back lobe – refers to a minor lobe that occupies the hemispheres in a direction opposite to that of the major lobe.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

93

Example of Radiation pattern lobes

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

94

8. More Antenna Parameters - Directivity, Beam Solid Angle, Gain, Antenna Equivalent Circuit.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

95

Directivity • A commonly used parameter to measure the overall ability of an antenna to direct radiated power in a given direction is the Directivity. • Directivity is defined as the ratio of radiation intensity in the direction of interest over the average radiation intensity.

U (θ , φ ) U (θ , φ ) U (θ , φ ) D(θ , φ ) = = Prad = 4π U average Prad 4π

(8.1)

• The maximum Directivity is usually denoted as Dmax or Do.

Do = D(θ , φ )max

U max = 4π Prad

(8.2)

• Directivity is a measure that describe the directional properties of an antenna, therefore it is controlled by the radiation pattern.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

96

Beam Solid Angle • Beam Solid Angle (ΩA) is the solid angle through which that all the power would be radiated if the radiation intensity U equals the maximum value over the beam area ΩA.

Prad = Ω AU max • Using (8.2) and (8.3): 4πU max

Do

= Ω AU max

(8.3) 4π ⇒ ΩA = Do

(8.4)

• Thus the larger the maximum Directivity, the smaller is the Beam Solid Angle of an antenna.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

97

Gain • Another useful measure describing the directional properties of an antenna is the Gain. It is defined as [2]:

G (θ , φ ) =

U (θ , φ ) Pin 4π

U (θ , φ ) = 4π Pin

(8.5)

where Pin = Total input (accepted) power into the antenna

• Note that there is a subtle difference between Gain and Directivity since Pin ≠ Prad as there is some power loss in a practical antenna.

U (θ , φ ) U (θ , φ ) G (θ , φ ) = 4π = 4π ≤ D(θ , φ ) Pin Prad + Ploss

(8.6)

• Under lossless condition the Gain and Directivity will be equal. Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

98

Radiation Resistance • A useful measure of the amount of power radiated by an antenna is the Radiation Resistance (Rrad). • The Radiation Resistance of an antenna is the value of a hypothetical resistance that would dissipate an amount of power equal to the radiated power Prad when the current in the resistance is equal to the maximum current along the antenna. • Naturally a high Radiation Resistance is a desirable property for an antenna.

Prad = ∫

2π

0

Oct 2009

∫

π

0

U (θ , φ )sin θdθdφ = 12 I m2 Rrad

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

(8.7)

99

Equivalent Circuit for Antenna • Usually the voltage across the antenna and the current into an antenna are not in phase. Thus we can model the antenna as a complex impedance at a particular frequency. RL

To account for loss in the antenna

jXA ZA = (RL+Rrad)+jXA Rrad

Account for electrical power being converted to radiation power

ZA = Antenna impedance

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

100

Example 8.1 • Find the Directivity, maximum Directivity, Beam Solid Angle and Radiation Resistance for the Hertzian Dipole.

U (θ , φ ) =

Prad = η=

µ ε

I o 2 ∆z 2ηk 2 (sin θ )2 32π

Directivity:

2

D(θ , φ ) = 4π

I o2 ∆z 2ηk 2 12π

≅ 120π

= In free space

12 (sin θ )2 32π

[

I o 2 ∆z 2ηk 2 (sin θ )2

⋅ 4π =

32π 2 3 2

Maximum Directivity and Beam Solid Angle:

Prad = 12 I o2 Rrad =

⇒ Rrad = Oct 2009

∆z 2ηk 2 6π

]

(sin θ )2

Do = D(θ , φ )max = Radiation Resistance

]/ [

I o 2 ∆z 2ηk 2 12π

I o2 ∆z 2ηk 2 12π

≅ 80π

2

3 2

8π ΩA = = 8.378 3

80π 2 2 ∆z 2 ≅ Io ( λ ) 2

[

[( ) ] ∆z 2

λ

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

]

Watt

Ohm 101

Exercise 8.1 • Find the Directivity, maximum Directivity, Beam Solid Angle and Radiation Resistance for the long Dipole (Half-wave dipole). Directivity:

GD (θ , φ )dipole = 4π U (θ , φ ) Prad = (

2 1.219

[

⇒ GD (θ , φ )dipole = 1.641 Max Directivity:

]

Max value is 1

Do = 1.641

Beam solid angle:

Radiation resistance:

Oct 2009

]

cos ( π2 cos (θ )) 2 sin θ

)[

cos ( π2 cos (θ )) 2 sin θ

4π ΩA = = 7.658 Ddipole Rrad

2 Prad 1.219ηk 2 = 2 = Im π

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

102

Polarization of an Antenna • In a given direction from the antenna, the polarization of the wave transmitted by the antenna. • Polarization of a wave describes the shape and locus of the tip of the E field vector at a given point in space as a function of time. • General locus is ellipse – elliptically polarized. • Under certain conditions – ellipse becomes a circle – circular polarization, or straight line – linear polarization.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

103

Polarization of a Wave • When E field is traced in clockwise direction – right-hand polarization, otherwise left-hand polarization. • Note that polarization rotation is opposite the direction of rotation of E field as a function of distance at a fixed point in time. • Common usage is with linear polarization, vertical and horizontal. • Both antenna and wave polarization must match for maximum power transfer.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

104

Polarization and Propagation Direction

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

105

Example of Polarization Mismatch Horizontally polarized E field

Vertically polarized E field

Take two horn antennas, one is horizontally polarised and the other is vertically polarised. Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

106

Example 8.2 – Classification of Radiation Pattern Omni-directional pattern

Omni-directional pattern

Omni-directional pattern

Directional pattern Directional pattern

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

107

9. Antenna in Receiving Mode and Reciprocity Theorem – Antenna Effective Area.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

108

Reciprocity Theorem for Antenna (1) • The discussion until now generally implied that the radiating system was to be used as a transmitting antenna, exciting EM waves in space from some high frequency energy source. • The same system useful for transmission are also useful for reception, and it will be seen that the quantities already calculated (such as the radiation pattern, directivity, antenna impedance for transmission) are also the useful parameters in the design of a receiving system. • In transmitting antenna, a generator is applied at localized terminals, and waves that are set up propagate in space approximately as spherical wavefronts. • In receiving antenna, a wave coming in from a distant transmitter approximates a portion of a uniform plane wave, and so sets up an applied electric field on the antenna system, quite different from that associated with the localized sources in the transmitting case.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

109

Reciprocity Theorem for Antenna (2) • Reciprocity Theorem for antenna systems provides the ties between the transmit and receive phenomena and it states that: – The antenna pattern for reception is identical to that for transmission. – The input impedance of the antenna on transmission is the internal impedance of the equivalent generator representing a receiving system. – An Effective Area (Ae) for the receiving antenna can be defined, and by reciprocity, is related to the Directivity. • Due to the scope of this subject, we are not able to go in depth into the proof of the above statements. The Reciprocity Theorem for antenna originates from the same theorem for EM fields in linear medium (see [6]). • The motivated student can refer to references [1], [2] or [4] for more details. Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

110

Reciprocity Theorem for Antenna (3) • The equivalent circuits for transmit (TX) and receive (RX) antenna. Uniform plane wave NOTE: Important concepts for Reciprocity Theorem for antenna: • Reciprocal devices – devices that exhibit the same radiation pattern for transmission as for reception. • Reciprocity may not hold for solid-state antenna composed of non-linear semiconductor devices or ferrite materials and active antenna. • Allows measurement to be made in either transmission mode or receiver mode, depending on convenience.

ZA

ZA + -

Oct 2009

Vs

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

111

Transmit and Receive System (1) • Based on Reciprocity Theorem, and using the equivalent circuit of an antenna, a transmit/receive (TX/RX) system can be modeled as shown. I1

I2

V1

V2

To load ZL2

RX

TX

I1

V1

I2

Linear 2-port network

ZL2

V2

• The relationships between the voltages and currents can be expressed using Z network parameters as:

V1 = Z11 I1 + Z12 I 2

V2 = Z 21 I1 + Z 22 I 2 Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

(9.1) 112

Transmit and Receive System (2) • The impedance Z12 is due to current induced in RX antenna causes EM wave that propagate back to the TX antenna. Usually this effect is small and we can ignore it. • Thus V ≅ Z I 1

11 1

V2 = Z 21 I1 + Z 22 I 2 I1

I2

V1

V2

To load ZL2

RX

TX I1

Z22

I2

+

V1

Oct 2009

Z11

I1Z21

ZL2

-

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

V2

113

Effective Aperture (Ae) (1) • From the equivalent circuit above for TX/RX system, the power 2 dissipated at load ZL is given by PL = 12 I 2 RL . • The power removed by the RX antenna can be thought of as an Effective Area (Ae) multiplied by the average Radiated Power Density at the antenna.

PR = S average Ae

(9.2)

• Ae provides an indication of the antenna ability to absorb the incident EM wave power density, and to deliver it to the load, thus the larger the better. • If we can measure the output power from an antenna in receive mode, and knowing the incoming EM wave power density, we can calculate the effective area using equation (9.2).

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

114

Effective Aperture (Ae) (2) • The role of TX and RX antenna can be reversed, thus each antenna has an effective area. I1

I2

V1

V2 Ae2

Forward condition TX To load ZL1

RX I2

I1 V1 RX

To load ZL2

V2 Reverse condition

Ae1

I1

TX I2

Z11 +

V1

Oct 2009

ZL1

I2Z12

Z22

-

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

V2

115

Ratio of Ae to D • By considering the equivalent circuit for ‘forward’ and ‘reverse’ condition and using the definition of Ae, we can show that, when the antenna is matched to the load (i.e. ZL2 = Z22*, ZL1 = Z11*):

Ae1 Ae 2 = D1 D2

(9.3)

• Where D1 and D2 are the respective Directivity or TX and RX antenna. • In general equation (9.3) applies to ALL antenna and the ratio of Effective Area to Directivity is a universal constant. By working out the ratio for a few antenna, it can be shown that:

Ae λ2 = D 4π

Oct 2009

(9.4)

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

116

10. Friis Transmission Formula for Transmit/Receive System

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

117

Power Captured by RX Antenna in TX/RX System • Consider a RX/TX system, separated by distance R. • Assume both antenna are at the direction for maximum sensitivity (e.g. max radiation intensity). R

I1

I2

V1

V2 RX

TX

PRX = Ae 2 S 21 S 21 = UR212

• Thus

PRX

Ae 2 Ae1 = 2 2 Pt1 Rλ

(10.1)

Radiated power density from antenna 1 (TX) to Transmit power from Antenna antenna 2 (RX). 1 (TX)

U 21 = D1

Radiation intensity from antenna 1 (TX) to antenna 2 (RX).

D1 = Oct 2009

To load ZL2

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

( ) Pt 1 4π

Directivity of Antenna 1

4πAe1

Effective Area of Antenna 1

λ2 118

Friis Transmission Formula • Another version of equation (10.1) in the form of Directivity is:

PRX

D2 D1λ2 = Pt1 2 (4πR )

(10.2)

• Both (10.1) and (10.2) are generally referred to as the Friis Transmission Formula.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

119

Example 10.1 • 100W is fed into a TX antenna, with the RX antenna at a distance of 10km away. Assume both TX and RX antenna as half-wavelength dipole, operating at a frequency of 879 MHz. – (a) Determine the received power at the RX antenna output. Assume both antenna to be impedance matched, pointed to each other at their maximum radiated intensity direction. – (b) Repeat part (a) if both antenna has a radiation efficiency of 90%.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

120

RADAR • A RADAR (Radio Detection and Ranging) is a TX/RX system in which the TX and RX antenna are at the same location and pointing towards the same direction. Sometimes the TX and RX antenna consist of just one antenna. Si =

Pt

TX RX

Pt DTX 2 4πR

R

Sr =

1 (Siσ s ) 2 4πR

Object RADAR cross section

Plane wave from TX antenna Oct 2009

Object ASSUMED to scatter the incident EM wave isotropically (D=1)

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

121

RADAR Equation and Cross Section • Assume both TX and RX are same type of antenna, or same antenna, thus DTX = DRX = D. • From (10.2) with D1 = 1 (for the object) and D2 = D, the receive power Pr at the RADAR is then: Pt D Dλ2 Dλ2 ( ) σ σ Pr = S = ⋅ ⋅ (4πR )2 i s (4πR )2 s 4πR 2

D 2 λ2σ s ⇒ Pr = P 3 4 t (4π ) R

(10.3)

• Equation (10.3) is called the RADAR Equation, it can also be taken as the definition for RADAR or Back-scatter Cross Section σs of an object. • The larger the back-scatter cross section of an object, the larger is the receive power at the RADAR.

Oct 2009

© Fabian Kung Wai Lee & Gobi Vetharatnam 2009

122