impedance and loop antennas - lanka education and research...

Impedance and Loop Antennas

Ranga Rodrigo

University of Moratuwa

January 4, 2009

Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, 2009 1 / 41

Summary of Last Week’s Lecture

Gain

Gain can be defined as

G =maximum radiation intensity

maximum radiation intensity from a ref-erence antenna with same power output

(1)

G0 =

maximum radiation intensity fromsubject antenna

radiation intensity from (lossless)isotropic source with same power input

(2)


Summary of Last Week’s Lecture

Gain

G = kD (3)

where k is the efficiency factor.


Summary of Last Week’s Lecture Efficiency

Efficiency

Efficiency =Pout

Pin=

Rrad

Rrad + Rloss. (4)


Summary of Last Week’s Lecture Antenna Aperture

Effective ApertureThe ratio of power W in the terminating impedanceto the power density of the incident wave will bedefined as the effective aperture Ae. Thus,

Effective aperture =WP

= Ae, (5)

where P = |P|. Thus,

Ae =V2RT

P[(Rrad + Rloss + RT)2 + (XA + XT)2

] . (6)

V is the induced voltage when the antenna isoriented for maximum response and the incidentwave has the same polarization as the antenna.


Summary of Last Week’s Lecture Maximum Effective Aperture

Maximum Effective Aperture

XT = −XA (7)

RT = Rrad. (8)

W′ =V2RT

4R2T

=V2

4Rrad. (9)


Summary of Last Week’s Lecture Maximum Effective Aperture

The power W′ is delivered to the terminatingimpedance under the conditions of maximumpower transfer and zero antenna losses.The ration of this power to the power density ofthe incident wave is the maximum effectiveaperture Aem.

Maximum effective aperture =W′

P= Aem. (10)

Aem =V2

4PRrad. (11)


Summary of Last Week’s Lecture Maximum Effective Aperture of a Hertzian Dipole

Maximum Effective Aperture of aHertzian Dipole

Aem =120πE2(dl)2λ2

320π2E2(dl)2 =3λ2

8π= 0.119λ2. (12)



Radiator Aem D D (dB)

Isotropicλ2

4π1 0

Short dipole1 3λ2

8π1.5 1.76

Half-wave dipole30λ2

73π1.64 2.14

1A short dipole is always of finite length even though it may be veryshort. The current along a short dipole is uniform.



Received Power in a CommunicationSystem

Pr =PtAetAer

R2λ2 . (13)

Pr = PtGtGr

(λ

4πR

)2

. (14)


Self and Mutual Impedances

Self and Mutual Impedances:Introduction

The impedance presented by an antenna to atransmission line can be represented by a2-terminal network.The impedance into which the transmission lineoperates is called the terminal and driving-pointimpedance.If the antenna is isolated, i.e., remote fromground and other objects, and is lossless, itsterminal impedance is is the same as theself-impedance of the antenna.



Antenna

Transmission line

ZEquivalent impedance



Self-impedance

Self-resistance

Self-reactance



The self-impedance is the same for reception asfor transmission.In the case there are nearby objects, theterminal impedance can still be replaced by a2-terminal network. However, its value isdetermined not only by the self-impedance butalso by the mutual impedance by it and the otherantennas and the currents flowing on them.


Self-Impedance of a Thin Linear Antenna

Self-Impedance of a Thin LinearAntenna

Assuming that the current distribution at theoutset to be nearly sinusoidal for dipoles withlength-diameter ratios as small as 100, and thatthe terminals are at a current maximum,self-impedance is given by

Z11 = R11 + jX11 Ω

= 30[Cin(2πn) + j Si(sπn)

]Ω

= 30[0.577 + ln(2πn) − Ci(2πn) + j Si(2πn)

]Ω.(15)



We will state what we mean by Cin, Ci, and Si now.They are three integrals. When we did the radiationresistance calculation, we avoided these integrals byusing the numerical result∫ π/2θ=0

cos2[(π/2) cosθ]sinθ dθ = 0.609.

Cin(x) =

∫ x

0

1 − cos(v)v

dv

= lnγx − Ci(x)= 0.577 + ln x − Ci(x).

(16)

where γ = ec = 1.781 or lnγ = c = 0.577 = Euler’sconstant.



Cosine IntegralThe part of this integral given by

Ci(x) = lnγx − Cin(x). (17)

is called the cosine integral. The value of thisintegral is given by

Ci(x) =

∫ x

∞

cos vv

= lnγx−x2

2!2+

x4

4!4−

x6

6!6+ · · · . (18)

When x is small (x < 0.2),

Ci(x) ' lnγx = 0.577 + ln x. (19)

When x is large (x 1),

Ci(x) =sin x

x. (20)Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, 2009 17 / 41


From 16 and 18 we obtain Cin(x) as an infiniteseries.

Cin(x) =x2

2!2−

x4

4!4+

x6

6!6− · · · . (21)



Sine IntegralThe sine integral is given by

Si(x) =

∫ x

0

sin vv

dv = x −x3

3!3+

x5

5!5− · · · . (22)


Si(x) ' x. (23)


Si(x) 'π2−

cos xx. (24)




Si(x) =

∫ x

0

sin vv

dv = x −x3

3!3+

x5

5!5− · · · . (22)


Si(x) '

x. (23)


Si(x) 'π2−

cos xx. (24)




Si(x) =

∫ x

0

sin vv

dv = x −x3

3!3+

x5

5!5− · · · . (22)


Si(x) ' x. (23)


Si(x) 'π2−

cos xx. (24)



-1

-0.5

0

0.5

1

1.5

2

0 2 4 6 8 10

Ci(x)Si(x)

Figure 1: Cosine and Sine Integrals



Self-Resistance and Self-Reactance

Self-Resistance

R11 = 30 Cin(2πn) Ω

= 30 [0.577 + ln(2πn) − Ci(2πn)] Ω.(25)

Self-Reactance

X11 = 30 Si(2πn) Ω. (26)



These equations give the impedance values fora thin linear center-fed antenna that is an oddnumber (n) of λ/2.

ExampleFind the self-impedance of a λ/2 antenna.

Solution

The antenna is a λ/2 antenna. Therefore, n = 1. Wehave for the self-resistance and self-reactance

R11 = 30 Cin(2π), (27)

andX11 = 30 Si(2π). (28)



-1

-0.5

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10

Ci(x)Si(x)

Cin(x)

Figure 2: Cin Integral



-1

-0.5

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10

Ci(x)Si(x)

Cin(x)

2π

2.44

Figure 3: Cin Integral



R11 = 30 Cin(2π)= 30 × 2.44 = 73 Ω.

(29)

We can show that, for a λ/2 dipole,

Z11 = R11 + jX11 = 73 + j42.5 Ω. (30)

Resonant “Half-Wave” DipoleSince X11 is not zero, an antenna exactly λ/2 long isnot resonant. To obtain a resonant antenna, it iscommon practice to shorten the antenna a fewpercent to make X11 = 0. In this case theself-resistance is somewhat less than 73 Ω.



R11 = 30 Cin(2π)= 30 × 2.44 = 73 Ω.

(29)


Z11 = R11 + jX11 =

73 + j42.5 Ω. (30)




R11 = 30 Cin(2π)= 30 × 2.44 = 73 Ω.

(29)


Z11 = R11 + jX11 = 73 + j42.5 Ω. (30)



Impedance at a Point Which is Not a Current Maximum

Impedance at a Point Which is Not aCurrent Maximum

When we use the methods above, what weobtain is the resistance and reactance at acurrent maximum.

This is not the point at which the transmissionline is connected for an antenna of generallength.



Impedance at a Point Which is Not aCurrent Maximum

When we use the methods above, what weobtain is the resistance and reactance at acurrent maximum.This is not the point at which the transmissionline is connected for an antenna of generallength.



Neglecting the antenna losses, the value ofradiation resistance so obtained is theresistance which would appear at the terminalsof a transmission line connected at a currentmaximum in the antenna, provided that theantenna current distribution on the antenna isthe same as when it is center-fed.

However, values can be transformed to thevalue which would appear across terminals ofthe transmission line connected at the center ofthe antenna.



Neglecting the antenna losses, the value ofradiation resistance so obtained is theresistance which would appear at the terminalsof a transmission line connected at a currentmaximum in the antenna, provided that theantenna current distribution on the antenna isthe same as when it is center-fed.However, values can be transformed to thevalue which would appear across terminals ofthe transmission line connected at the center ofthe antenna.



I0I1

x

A 3λ/4 dipole.

I=I0 cos βx. (31)

where

I1 = terminal current,I0 = maximum current.

This may be done byequating Prad = I2

0Rrad/2 tothe power supplied by thetransmission line I2

1R1/2,where I1 is the currentamplitude at the terminalsand R1 is the radiationresistance at this point.

I21

2R1 =

I20

2Rrad. (32)



We can write

R1 =(I0

I1

)2

R0. (33)

Therefore,

R1 =Rrad

cos2 βx. (34)

The reactance X1 is given by

X1 =Xrad

sin2 βx. (35)

where Xrad is the reactance seen at a currentmaximum point.


Loop Antennas

Loop Antennas

Loop antennas are simple, inexpensive and veryversatile.They may be rectangular, circular, triangular orelliptical.Small loop antennas are defined as those withcircumference less thanλ/10 whereas largeloops are those with circumference on the orderof λ.They are mostly used in HF, VHF and UHFranges and are commonly used as probes atmicrowave frequencies.


Loop Antennas

Small Loop AntennasSmall loop antennas are usually used asreceivers rather than transmitters since they arevery compact, but not effective in transmittinglarge amount of power.Portable radios and pagers, where antennaefficiency is not very important as the signal tonoise ratio, utilize small loops as receivers.They are also used as probes for fieldmeasurement as well as for directional antennasin radio navigation.Reception can be increased by increasing theperimeter, the number of turns, or by inserting aferrite core (ferrite loop antenna).


Loop Antennas

The Small Loop AntennaThe field pattern of a small circular loop of radiusa may be determined very simply by consideringthe square loop of the same area, that is,

d2 = πa2 (36)

where d is the side length of square loop.It is assumed that the loop dimensions are smallcompare to the wavelenght.The far-field patterns of circular and squareloops of the same area are the same when theloops are small but different when they are largein terms of the wavelenght.


Loop Antennas

2a d

Figure 4: Circular and Square Loops of Equal Area.


Loop Antennas

z

y

x

12

34

Figure 5: Orientation of the Square Loop.


Loop Antennas

z

y

θ

To distant point

d

Figure 6: Construction for Finding the Far Field of Dipoles 2and 4 of Square Loop.


Loop Antennas

If the loop is oriented with its plane lying in the xyplane, its far electric field has only an Eφ component.To find the far-field pattern in xy plane, it is onlynecessary to consider two of the four small lineardipoles (2 and 4). Since the individual small dipoles2 and 4 are nondirectional in the yz plane the fieldpattern of the loop in this plane is the same as thatfor two isotropic point sources.

Eφ = −Eφ0ejψ/2 + Eφ0e−jψ/2, (37)

where Eφ0 is the electric field from the individualdipole and

ψ =2πdλ

sinθ = dr sinθ, where dr =2πdλ. (38)


Loop Antennas

Eφ = −2jEφ0 sin(dr

2sinθ

). (39)

If d λ 38 can be written

Eφ = −jEφ0dr sinθ. (40)

Eφ0 is the electric field due to an individual dipole.These dipoles are short dipoles. We saw that thefar-field pattern due to a Hertzian dipole oriented inthe z-direction is

E = −ηβI0dl sinθ

4πrsin(ωt − βr)iθ. (41)

where β = 2πλ and η = 120π. A similar formula exists

for the short dipole.Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, 2009 37 / 41

Loop Antennas

The far-field pattern due a short dipole is

E =jηβI0L sinθ

4πrsin(ωt − βr)iθ

Eθ =jηβ[I]L sinθ

4πr

=j60π[I] sinθ

rLλ.

(42)

where L is the length of the dipole, and[I] = I0 sin(ωt − βr), the retarded current.


Loop Antennas

The angle θ in the dipole formula is measured fromthe dipole axis and is 90 in the present case. Theangle θ is a different angle with respect to the dipole.Thus we have for the far field Eφ0 of the individualdipole

Eφ0 =j60π[I]L

rλ(43)

Now,

Eφ =60π[I]Ldr sinθ

rλ(44)

Length of the dipole L = d, dr = 2πdλ , the area of the

loop is A = d2.


Loop Antennas

Far Field of a Small Loop

Finally, we have

Eφ =120π2[I] sinθ

rAλ2 . (45)

Hθ =Eφ

120π=π[I] sinθ

rAλ2 . (46)


Loop Antennas

ReferenceIndra J. Dayawansa.Antennas and propagation.Lecture notes, University of Moratuwa, 2003.

John D. Kraus.Antennas.McGraw-Hill, 1950.

John D. Kraus, Ronaled J. Marhefka, and Ahmad S. Khan.Antennas for All Applications.Tata-McGraw-Hill, 3rd edition, 2006.

Nannapaneni Narayana Rao.Elements of Engineering Electromaganetics.Prentice Hall, 4th edition, 1994.Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, 2009 41 / 41

impedance and loop antennas - lanka education and research...

Documents