P5-Electromagnetic Fields and Waves · P5-Electromagnetic Fields and Waves ... •Appreciate when a wave theory is needed ... Download a PDF of these slides from

1SPECTROSCOPY GROUPNANOMATERIALS AND CAMBRIDGE UNIVERSITY

DEPARTMENT OF ENGINEERINGELECTRONIC DEVICES

AND MATERIALS GROUP

P5-Electromagnetic Fields and Waves

Andrea C. Ferrarihttp://www-g.eng.cam.ac.uk/nms/lecturenotes.html

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AND MATERIALS GROUP

6 Lectures3 Main Sections ~2 lectures per subject

I Transmission Lines

I.1 Telegrapher’s EquationsI.2 Characteristic ImpedanceI.3 Reflection

I.0 The wave equation

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II Electromagnetic Waves in Free SpaceII.1 Electromagnetic FieldsII.2 Electromagnetic WavesII.3 Reflection and Refraction of Waves

III Antennae and Radio TransmissionIII.1 Antennae

III.2 Radio

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OBJECTIVESAs the frequency of electronic circuits rises, one can no longer assume that voltages and currents are instantly transmitted by a wire.

•Appreciate when a wave theory is needed

•Derive and solve simple transmission line problems•Understand the importance of matching to the characteristic impedance of a transmission cable

•Understand basic principles of EM wave propagation in free space, across interfaces and the use of antennae

The objectives of this course are:

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Reference: OLVER A.D. Microwave and Optical TransmissionJohn Wiley & Sons, 1992, 1997

Shelf Mark: NV 135

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Handouts

Left-hand Column: Read if you need extra clarification

The handouts have some gaps for you to fill

In both CASES 1) and 2) You will be able to Download a PDF of these slides from

http://www-g.eng.cam.ac.uk/nms/lecturenotes.html6

1) DO NOT PANIC IF YOU DO NOT MANAGE TO WRITE DOWN IN “REAL TIME”

2) Prefer to just sit back and relax?

NOTE:



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I.0 The Wave Equation

AimsTo recall basic phasors conceptsTo introduce the generalised form of the Wave Equation

ObjectivesAt the end of this section you should be able to recognisethe generalised form of the wave equation, its general solution, the propagation direction and velocity

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I.0.0 IntroductionAn ideal transmission line is defined to be a link between two points in which the signal at any point equals the initiating signal

I.e. transmission takes place instantaneously and there is no attenuation

Real world transmission lines are not ideal, there is attenuation and there are delays in transmission

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A transmission line can be seen as a device for propagating energy from one point to another

The propagation of energy is for one of two general reasons:

1. Power transfer (e.g. for lighting, heating, performing work) - examples are mains electricity, microwave guides in a microwave oven, a fibre-optic illuminator.

2. Information transfer. examples are telephone, radio, and fibre-optic links (in each case the energy propagating down the transmission line is modulated in some way).

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PowerPlant

ConsumerHome

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Examples



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CoCo--axax cablecablePair of wiresPair of wires

PCB tracksPCB tracks IC interconnectsIC interconnects

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Waveguides

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Dielectric of thickness T, with a conductor deposited on the bottom surface, and a strip of conductor of width W on the top surface

Mircostrip

Can be fabricated using Printed Circuit Board (PCB) technology, and is used to convey microwave frequency signals 14



AND MATERIALS GROUP 15Microwave Oven



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Optical Fibres

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Phasor Notation

A means A is complex

{ } { } i AA e A m A i A e

A A

m

e{ }e A

AA{ }m A

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j xAe is short-hand for ( )e j x tAe

which equals: cosA x t A

Proof

cos( ) sin( )ie j

cos( ) sin( )j x t AAe A x t A jA x t A

then

( ) ( ) ( )j x t j A j x t j x t AAe Ae e Ae

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I.0.1 The Wave EquationThe generalised form of the wave equation is:

22 2

2

A v At

Where the Laplacian of a scalar A is:

2 2 22

2 2 2

A A AAx y z

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We will be looking at plane waves for which the wave equation is one-dimensional and appears as follows:

2 22

2 2

A Avt x

Where A could be:

Either the Voltage (V) or the Current (I)as in waves in a transmission line

Or the Electric Field (E) or Magnetic Field (H)as in Electromagnetic Waves in free space

or2 2

2 2 2

1A Ax v t

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The constant v is called the wave speed, from the general solution to the wave equation:

( )A f x vt

Note

( )A f x vt Forward moving

( )A f x vt Backward moving 21



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ΔΔxx

xx

ΔΔxx

t+t+ΔΔt tt t

F(t+x/vF(t+x/v))

xx

If t increases (t→t+∆t), x must also increase if x-vt is to be constant

Direction of travel

Consider a fixed point, P, on the moving waveform, i.e. a point with constant f

x increases implies wave is moving to right (Forward)

f(x-vt) will be constant if x-vt is constant

PPt t+t t+ΔΔtt

Similarly, for x+vt wave is moving to left (Backward)

( )f x vt( )f x vt

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Proof f is general solution

( )A f x vt

'( )A vf x vtt

22

2 ''( )A v f x vtt

'( )A f x vtx

2

2 ''( )A f x vtx

2 22

2 2

A Avt x

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I.1 Electrical Waves

AimsTo derive the telegrapher’s equationTo account for losses in transmission lines

ObjectivesAt the end of this section you should be able to recognisewhen the wave theory is relevant. To master the concepts of wavelenght, wave velocity, period and phase. To describe the propagation of waves in loss-less and lossytransmission lines

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L

I.1.1 Telegrapher’s EquationsLet us consider a short length of a wire pairLet us define

series/loop inductance per unit length [H/m]

LV

I

LIV L xt

x

L x

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C parallel/shunt capacitance per unit length [F/m]

CI

CC

VI C xt

CV

x

C x

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C LV V V

FII I xx

x

CVV V xx

CC

VI C xt

LIV L xt

I

CII xx

CVLV

A

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C LV V V V IV x V L xx t

F CI I I I VI x I C xx t

V ILx t

I VCx t

(1.1)

(1.2)

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I.1.2 Travelling Wave Equations

2

2

V Lx t

Ix

Ix

2

2

I Cx t

Vx

(1.1a)

(1.2a)

Then in (1.1a) substitute

Let us differentiate both (1.1) and (1.2) with respect to x

using (1.2)

Then in (1.2a) substitute Vx

2

2

VLCt

2

2

ILCt

using (1.1)29



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2 2

2 2

V VLCx t

2 2

2 2 2

1A Ax v t

2 2

2 2

I ILCx t

(1.1a)

(1.2a)

Same functional form as wave equation:

We try a solution for V in (1.1a) of the formj x j tV Ae e

2 1vLC

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2 2j x j t j x j tAe e LC Ae e

LC

j x j x j tF BV e V e V e e

j x j x j tF BI e I e I e e

Hence

Phase Constant (1.3)

Since β can be positive of negative we obtain expressions for voltage and current waves moving forward (subscript F) and backward (subscript B) along the transmission line

(1.4)

(1.5)

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I.1.3 Lossy Transmission LinesOur expressions have been derived for a lossless transmission line and therefore do not include resistance along the line or conductance across the line

To derive the relevant expressions for a lossy transmission line our equivalent circuit would become :

R= Series resistance per unit length [/m]

G= Shunt conductance per unit length [S/m]

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A

x

0R L CV V V V

G x C x

R x

V

L xI

II xx

CVV V xx

RV LV

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( ) 0I VV R xI L x V xt x

( ) j tI f x e

( )V R j L Ix

V IRI Lx t

For simplicity we assume

( ) j tI j f x e j It

Then

Compare with (1.1) V ILx t

j LI

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Similarly using Kirchoff’s current law to sum currents:

( ) 0II G xV j C xV I xx

( )I G j C Vx

I VC j CVx t

Compare with (1.2)

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L in a lossless line corresponds to:

( )' G j CCj

( )' R j LLj

in a lossy line

C in a lossless line corresponds to:

in a lossy line

Then LC In a lossless line corresponds to:

1' ( )( )R j L G j Cj

in a lossy line

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We get (1.6)j x j x j t

F BV e V e V e e (1.7)j x j x j t

F BI e I e I e e

Substituting β β’ into (1.4) and (1.5) and defining

( )( )R j L G j C j

is the phase constant

And the real term α corresponds to the attenuation along the line and is known as the attenuation constant.

is called propagation constant

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VoltageVF

x

For a forward travelling wave:

V = VF e jt e-x = VF e-αx e j(t-βx)

amplitude factor phase factortime variation

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0

2( )( )R j L G j C LC j LC

L R :C G At high frequencies, where

The expressions approximate back to those for lossless lines

and

Note:

Thus

LC

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I.1.4 Wave velocity: vOur expressions for voltage and current contain 2 exponentials

The one in terms of x: j xe

Gives the spatial dependence of the wave, hence the wavelength:

2

The other: j te

gives the temporal dependence of the wave, hence its frequency:

2f

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For a wave velocity v, wavelength and frequency f:

v f

22

v

then

LC since

1vLC

(1.8)

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I.1.5 Example: Wavelength An Ethernet Cable has L= 0.22 μHm-1 and C = 86 pFm-1.What is the wavelength at 10 MHz ?

2

LC From

23metres

And

2LC

Then 6 6 12

22 *10*10 0.22*10 *86*10

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I.1.6 When must distances be accounted for in AC circuits?

2

4

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16L

16L

Wave Relevant

Wave Not Relevant

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I.1.7 Example: When is wave theory relevant?A designer is creating a circuit which has a clock rate of 5MHz and has 200mm long tracks for which the inductance (L) and capacitance (C) per unit length are:

L=0.5Hm-1 C=60pFm-1

2

LC From And 2LC

6 6 12

2 36.52 *5*10 0.5*10 *60*10

m

Then

36.5 m is much greater than 200 mm (the size of the circuit board) so that wave theory is irrelevant.

Note: The problem is even less relevant for mains frequencies i.e. 50 Hz. This gives ~3650 km 45

Documents

P5-Electromagnetic Fields and Waves · P5-Electromagnetic Fields and Waves ... •Appreciate when a wave theory is needed ... Download a PDF of these slides from