expt_7

Experiment 7. Wave Propagation

Updated RWH 21 August 2012

1 Aim

In this experiment you will measure the radiation pattern ofa half-wave dipole antenna, deter-mine the resonant frequencies of a microwave cavity, and examine standing waves in a rectan-gular waveguide.

Useful general references for this experiment are Cheng [1], and Ramo et al [2].

2 Radiation from a half-wave antenna

Antennas are structures designed for radiating and receiving electromagnetic energy in pre-scribed patterns. These patterns depend on the geometry of the antenna and can take manydifferent forms. The radiation originating from a linear dipole antenna depends critically onits length. The radiation pattern of a half-wave antenna (one whose length is half a radiationwavelength) is studied in this section. A reflector element is then added to enhance the signalin one preferred direction.

7–2 SENIOR PHYSICS LABORATORY

2.1 Measuring the radiation pattern

From the theory of radiating antennas one can deduce the expected polar dependence for a half-wave dipole. The radiation intensity (defined as the time average power per unit solid angle)can be expressed as follows:

U = Umax

∣

∣

∣

∣

cos(π2

cos θ)

sin θ

∣

∣

∣

∣

2

(1)

whereθ is the angle between a line along the prongs of the antenna anda line joining the middleof the antenna to the detector; see Fig. 7-1.

To record the radiated intensity as a function of angle,θ, the signal generator is coupled to thedipole antenna through a rotating joint. The fixed receivingantenna also has the same lineardipole geometry as the transmitter and incorporates a built-in diode detector. The output fromthe detector is fed via a coaxial cable to a sensitive DC current meter whose output is fed to adigital voltmeter (DVM). You can read the output signal either from the analogue meter scaleor the DVM.

The diode detector has a “square law” dependence on output with respect to the input, i.e, thetime averaged forward current is proportional to the squareof the input voltage, and since thepower is also proportional to voltage squared (P = V 2/R), the measured output current isproportional to the received power.

Question 1:We assume that the detector diode responds according to the Shockley modelI–Vcharacteristic:

I = Is[exp(eV/nkT ) − 1], (2)

wheree is the electron charge,V is the input voltage,n is known as the diode ideality factor(for silicon diodesn ≈ 1–2),k is Boltzmann’s constant,T is the absolute temperature,Is is theinherent leakage current of the diode (whenV has a large negative value) andI is the measuredforward current. Assuming that the input voltage is small and has the formV = V0 sinωt, andthat the meter records the average current (because the frequency is so high), prove that thetime averaged forward current is proportional to the squareof the input voltage.Hint: Start by expanding Eq. 2 in a Taylor series, assuming thatV0 ≪ nkT/e, wherenkT/e ≈30 mV at room temperature.

Procedure

1. Install the transmitting antenna on the output of the microwave generator using a directconnection through a rotating joint.

2. Set the generator’s frequency control to 280 (i.e.∼3.0 GHz).

3. Set the transmitting and receiving antennas about 40–50 cm apart and at the same height.

4. Make sure that they are both at right angles to the line joining them, using the plastictube holding the receiving antenna for alignment.

5. Perform a test to check that the radiation is plane polarised by rotating the receivingantenna about the axis of its plug and noting the changes in the reading of the detector

WAVE PROPAGATION 7–3

current1. The maximum reading should occur when the receiving antenna lies on thesame plane as that formed by the transmitting dipole.

6. Take readings of the detector current (or DVM voltage) every 10◦ as the transmitterantenna is rotated through 360◦, as shown in Fig. 7-1. Plot the polar diagram of thetransmitting antenna (i.e., received power versus angle) using either Radar plot in Excelor a similar function in Open Office.

Fig. 7-1 Plan view (schematic) for measuring the directional characteristic of the transmitting antenna.

7. IdentifyUmax from your experimental data (corresponding toθ = 90◦),

8. Tabulate the expected theoretical response for each angle and plot in a different colouron the same polar diagram as the experimental data.

9. Compare both diagrams and comment.

C 1 ⊲

2.2 Radiation with a reflector element

It is often desirable to concentrate the radiation from an antenna in a particular direction. Byadding one or more “parasitic” elements we can improve the front-to-back ratio and henceincrease the directionality of the antenna array.

By introducing a reflector element behind the transmitting antenna we can maximise the signalin the forward direction. A current is induced in the parasitic element by the current in thedriven element. Its phase lag will depend on the distanced between them and the lengthlof the reflector. Ifl > λ/2 there is a phase change of180◦ on reflection. The field radiatedfrom the antenna will be the vector sum of the fields from each element, so by selecting thepath difference appropriately one can achieve constructive interference in one direction anddestructive interference in the opposite direction.

1The nanoammeter should be zeroed only on the 0.3µA range by pressing thePRESS to set zero button andadjusting thezero knob. It’s a good idea to check the zero during the measurement sequence.


The interference between elements to improve directionality and to enhance the signal in apreferred direction works just as well with a receiving antenna. This is exploited in the designof the ubiquitous Yagi-Uda antennas found on top of most houses for TV or FM reception. Aswell as a single reflector element (as studied above) these antennas use many director elementsin front of the “driven” element which is connected to the TV or FM receiver. Only one reflectorelement (usually slightly longer in length than the driven element) is needed. However, byoptimising the number and separation of the somewhat shorter director elements in front of thedriven element, the directionality (and bandwidth) can be markedly improved. The antenna isnow sensitive to radiation coming in a narrower cone about its axis, compared to a single dipoleor a single dipole plus reflector.

The radiation intensity for a half wave dipole transmitter separated by a distanced from areflector element withl > λ/2 is given by:

U = Umax

∣

∣

∣

∣

cos(π2

cos θ)

sin θ

∣

∣

∣

∣

2∣

∣

∣

∣

cosψ

2

∣

∣

∣

∣

2

(3)

whereψ = kd(sin θ + 1) − π is the phase factor andk = 2π/λ is the wave number.

Question 2: Ideally, how far apart should the two elements be to achieve constructive interfer-ence in the forward direction and destructive in the backward direction?

Procedure

1. Check the validity of your prediction in Question 2 by adjusting the position of the re-flector element to find the first two maxima and the first minimum. Measure the distancesbetween transmitter and reflector elements for these three cases.

2. Comment on your observations and on any anomalies you may find. (At smaller dis-tances the mutual impedance between the two elements becomes more important andintroduces an extra phase term dependent on the distance between the transmitter andreflector. See Wolff, pages 296–298.)

3. By placing the reflector at the position of maximum forwardsignal, plot the polar dia-gram in this configuration and comment on the signal enhancement in the forward direc-tion.

4. Compare your experimentally measured results with calculations of the expected re-sponse using Eq. 3 by plotting them on the same polar diagram.

5. How has the directionality of the antenna improved in comparison to the dipole withouta reflector?

C 2 ⊲

3 Cylindrical resonant cavity

A cylindrical cavity can be visualised as a circular waveguide with two conducting ends on it.Resonant modes for these cavities can be either transverse magnetic (TM) or transverse electric


(TE) and are defined by three indices, i.e.TMmnp and TEmnp. The first indexm gives thenumber of cycles as the azimuthal coordinateφ goes through360◦. The second indexn givesthe corresponding variation with radius from the guide axisto the conducting wall. The thirdindexp gives the number of cycles in the axial (z) direction.

There is an interesting dependence of the electric and magnetic fields on radius from the axis.For theTM modes the radial component of the magnetic field, the azimuthal and axial compo-nents of the electric field all depend on radius as Bessel functions of integral orderJm. Theazimuthal component of the magnetic field and the radial component of the electric field de-pend on radius as the first derivative of the Bessel functionJ ′

m . ForTE modes, swap the wordselectric and magnetic. For both sets of modes the boundary conditions must be obeyed: theelectric field must be normal to the walls and the magnetic field tangential. If the radius of thecavity isa and the heighth, then theTMmnp modes are given by:

Er = AJ ′

m(kcr) cos(mφ) sin(πpz/h) (4)

Eφ = (B/r)Jm(kcr) sin(mφ) sin(πpz/h) (5)

Ez = CJm(kcr) cos(mφ)cos(πpz/h) (6)

with Jm(kcr) = 0 at r = a.

TheTEmnp modes are given by :

Er = (D/r)Jm(kcr) sin(mφ) sin(πpz/h) (7)

Eφ = FJ ′

m(kcr) cos(mφ) sin(πpz/h) (8)

Ez = 0 (9)

with J ′

m(kcr) = 0 at r = a. A,B,C,D,F are constants depending on the wave amplitude,andk2

c = (ω/c)2 − k2z , with kz = 2π/λz andpλz = 2h.

We havesin(πpz/h) = 0 at the ends whilecos(πpz/h) is a maximum there. Depending onwhether the mode isTM or TE as given above,Jm or its derivativeJ ′

m will have a zero at thecavity wall r = a. In general, this will not be the first such zero (the indexn gives the orderof the zero) but for the modes you will be asked to investigaten = 1 (i.e. the first zero is atr = a). There is more information in the reference books by Gardiol [4] and Ramo et al [2].

Each of the different modes of vibration for a cavity of a certain size can be associated with aresonant frequency. At resonance the amount of energy absorbed by the cavity is a maximum.

Question 3:Prove that the resonant wavelengths associated with the first three resonant modesof oscillation are as follows:

λ0 = 2.61a for the TM010 mode

λ0 =2h

√

1 + (2h/3.41a)2for the TE111 mode

λ0 =2h

√

1 + (2h/2.61a)2for the TM011 mode

To do this you will need to make the parallel component of the electric field zero at the wallsof the cavity, knowing that:

J0(kca) = 0 when kca = 2.405 and

J ′

1(kca) = 0 when kca = 1.841.


Note thatk2 = k2c + k2

z , wherekc is the transverse component andkz is the axial componentof the propagation vector, and the resonant wavelength isλ0 = 2π/k.

We can affect the electric field inside the cavity by inserting conducting “needles” throughstrategically placed holes drilled through the sides or ends. If the insertion of a needle producesa shift in the resonant frequency it must mean that there is a component of electric field parallelto the direction of the needle.

Fig. 7-2 Experimental arrangement for cavity measurements.

Procedure

1. Connect the apparatus as shown in Fig. 7-2. We use the BWD function generator tosweep the microwave generator with a 1 kHz asymmetrical triangular wave (SYMM andtriangle wave buttons pressed).

2. Connect the labelled lead to the X channel (CH1) of the Agilent DSO1002A digital os-cilloscope, and connect the output from the detector to the Ychannel (CH2). Make surethe oscilloscope is in X-Y mode, and the sampling rate is set to 125k samples/second.Resonances will appear as sharp dips on the trace as the frequency setting is adjustedupwards from a setting of 100 on the dial. Disregard the more numerous small “bumps”;they are due to weak emissions from the microwave generator at 2f , 3f , etc..

3. Measure the first three resonant frequencies accurately by progressively reducing thesweep amplitude so that the resonant dip is centred. You willneed to interpolate betweenadjacent entries in the calibration table to determine the central frequency or, better still,run a cable to the frequency counter on the Microwaves experiment.

4. Identify the following modes:TM010, TE111 andTM011 using a comparison of theoreticaland measured resonant frequencies, and confirm by describing the results of using the


needle technique to trace the field patterns. The cavity radius isa = 4.902cm and itsheight ish = 7.646cm.

5. Using the dependence of the different components of the electric field shown in Eqns4–9, make neat sketches of the electric field patterns for thethree resonant modesTM010,TE111 andTM011. A diagram of the first four Bessel functions is shown in Fig. 7-3.

+1

0.5

0

-0.5

J0

J1

J2 J 3

J

4

x

NB J 0 (x) = -J 1 (x)

1 2 3

4

5 6

7 8 9

Fig. 7-3 Dependence of Bessel functionsJn(x) as a function ofx.

C 3 ⊲

4 TE10 wave in a rectangular guide

A waveguide is a structure that causes a wave to propagate in achosen direction while confin-ing it inside a prescribed geometry. An open-ended rectangular waveguide will be studied inthis section. The metal boundaries impose the condition that the tangential component of theelectric field remains zero and thus thex andy axes support standing waves. Even though thenet propagation of energy only occurs along thez axis, we also find standing waves along thisaxis because the waveguide is not “matched” (the intrinsic impedance of the waveguide is notthe same as the impedance of free space).

By analogy with the cylindrical cavity, the allowed modes ofoscillation in a rectangular waveg-uide areTMmn andTEmn, whereTM means that the magnetic field is entirely transverse (per-pendicular to thez axis) andTE that the electric field is transverse. The first indexm givesthe number of cycles of variation of the field allowed along the x axis while the secondn de-scribes the number of cycles allowed along they axis. By choosing the frequency of operationappropriately we can select just theTE10 oscillation mode.

There is a linear relationship between the inverse of the square of the free-space wavelength(λ) and the inverse of the square of the guide wavelength (λz), given by

1

λ2=

1

λ2z

+1

4a2(10)

wherea is the guide width in thex direction.λc = 2a is called the cut-off wavelength becausethis is the maximum wavelength that the guide can support (for theTE10 mode).


Wave Guide

to DC Amplifier UHF to NAdaptor

DetectorConicalAntenna

PlasticBox

MicrowaveGenerator

N toBNCAdaptor

Carriage

Fig. 7-4 Experimental arrangement for waveguide measurements.

Procedure

1. Connect the microwave generator to the conical antenna atthe end of the waveguideprovided. Power picked up by the sliding probe passes through an adaptor into a detectorand then to the DC nanoammeter (see Fig. 7-4).

2. Starting with the frequency control set to 100, move the probe carriage back and forth tofind the positions (all of them) where the DC nanoammeter reads a minimum. Becausethese minima are generated by standing waves in thez direction, the average separationbetween successive minima is equal to half the guide wavelength, i.e.,λz/2.

3. Repeat this for the following settings of the generator’sfrequency control: 150, 200, 250,300, 400, 500, 600 and 700.

4. For each frequency setting, read the free-space wavelength from the calibration table (ormake a direct frequency measurement using the Microwaves counter) and plot a graphof 1/λ2 against1/λ2

z .

5. Determine the slope of the graph. Is it consistent with unity, as suggested by Eq. 10? Usethe intercept to calculate the cut-off wavelength. Compareyour estimate of the width ofthe waveguide with a direct measurement made with a vernier caliper.

C 4 ⊲


References

[1] David K. Cheng,Field and wave electromagnetics, Addison-Wesley, Reading, Mass,1992.

[2] S. Ramo, J.R. Whinnery and T. van Duzer,Fields and waves in communication electron-ics, Wiley, New York, 1994.

[3] E.A. Wolff, Antenna Analysis, Artech House, Norwood, MA, 1988.

[4] F.E. Gardiol,Introduction to Microwaves, Artech House, Dedham, MA, 1984.

[5] E.L. Ginzton.Microwave Measurements. McGraw Hill, NewYork, 1957.

[6] Reference Data for Radio Engineers, Howard W. Sams and Co. Inc.

expt_7

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