**RELATIONSHIP BETWEEN ****λ**_{g }and **λ**_{a}

**Objective****: ** To provide the relation

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where

λ_{a} is the wavelength in free space

λ_{g} is the wavelength in waveguide

a is he broad dimension of the waveguide.

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**List of Equipments:**

1. Microwave source with square wave modulation

2. Isolator

3. Variable attenuator

4. Crystal detector

5. VSWR Meter

6. Ferrite Isolator

7. Horn antenna

8. Reflecting sheet

**Theory: **

The best way to prove this relationship is to measure λ_{g} and λ_{a} and then plot the values of (1/λ_{a})^2

against the corresponding values of (1/λ_{g})^2. This should result in a straight line graph which cuts the (1/λ_{a})^2axis at a value equal to (1/2a_{})^2. If this graph allows the determination of a value of width of the broad wall of the guide which agrees with the actual value as measured by Calipers then this surely is a proof of the validity of the relationship expressed in the above equation.

To measure the waveguide wavelength λ_{g} , standing waves are produced by improperly terminating the waveguide. Secondly, the radiation is allowed to leave the waveguide and the wavelength of the standing wave in free space (λ_{a}) _{is }compared with the wavelength in the waveguide(λ_{g}). To produce standing wave in the waveguide and in the air, a horn is attached to the waveguide and a sheet of metal is placed a certain distance way from the horn. The electromagnetic radiation will then leave the waveguidevia the horn into the air. It will be reflected from the sheet of metal and produce a standing wave between the horn and the sheet with a wavelength, λ_{a}. As the reflected wave will re-enter the waveguide there will be a standing wave produced in the waveguide but with a wavelength λ_{g}.

If the sheet is moved relative to the horn keeping its plane at right angles to the axis of the waveguide the standing wave will move. Therefore, the signal at the stationary probe of the standing wave detector will vary as the standing wave moves past it. The distance moved by the sheet to repeat the output on the meter (i.e. to have the probe at the ext corresponding position on the standing wave) is half a free space wavelength λ_{a}/2. The voltage of the sanding wave must always be at a minimum at the sheet because it is a good conductor. Consequently, moving the sheet moves the pattern. Therefore, when the sheet has moved a distance λ_{a}/2, the voltage at the stationary probe of the standing wave detector will be repeated. A measurement of the amount of movement of the sheet determination λ_{a}/2 at particular frequency.

To determine the guide wavelength λ_{g}/2 at the same frequency all that is necessary is to keep the sheet still and move the standing wave detector probe. The pattern will remain stationary and the probe will explore that pattern in the waveguide which repeats every half guide wavelength λ_{g}/2. Recording the distance between two adjacent minima by noting the reading on the scale of the standing wave detector gives λ_{g}/2.

Now knowing λ_{g}/2 and λ_{a}/2 at one frequency, these values are recorded and the whole experiment is repeated at a new frequency. This procedure is repeated for measurement at a number of frequencies. The values of (1/λ_{g})^2 and (1/λ_{a})^2 for each frequency are plotted and from the intersection of the graph with the (1/λ_{a})^2 axis a value of the inside broad dimension of the waveguide can be obtained, and compared with the measured value.

**Procedure: **

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**Fig.1 Experimental arrangement for measuring λ**_{g}** and ** **λ**_{a}

1. Set the frequency and amplitude of the source. Fix the modulator frequency using slider. Select “**Forward direction**” in isolator and adjust the attenuation using slider.

2. To produce standing wave pattern we use waveguide horn and metal sheet. First we will calculate λ_{a} (free space wavelength). For that fix the “**Resistance**” and “**Reactance**” of antenna by using slider. Now move the metal sheet by using slider. Move slider of slotted line to get the minimum output in free space. Then move the slider to get the minimum value as obtained before.

λ_{a}/2 = distance between two same minimum value obtained in free space

3. Press “**First free space minima**” & “**Second free space minima**” to record the corresponding distance.

4. To find the guided wavelength λ_{g} move the slider in waveguide to get the positions of minima. In simulation slider works as a probe in case of slotted line.

λ_{g}/2 = distance between two minima obtained in slotted line

5. Now fix those two values by pressing two button “**First waveguide minima**” & “**Second waveguide minima**”.

6. Press “**Width of Waveguide**” to know the value.

a = Width of the waveguide

7. Change the frequency in small steps, say 200 MHz. Measure the corresponding λ_{a}and λ_{g}. Do this to cover a frequency range of, say, 9-11 GHz.

8. Plot a graph of (1/λ_{g})^2 (y-axis) vs. (1/λ_{a})^2 (x-axis). The intersection of the graph with x-axis equals, (1/2a)^{2},enabling one to arrive at an estimate of a. This estimate may be compared with the actual value

**Discussions: **

1. Why metal sheet is used? What effect does it produce?

2. Why horn antenna is used? Can we use any other antenna?

3. What do you mean by guided wavelength?

**References **

1. S. Ramo, J.R. Whinney and I. VanDuzer, Fields and Waves in Communication Electronics, Third Edition, John Wiley & Sons, 1994.

2. E.L. Ginzton: Microwave Measurements, McGraw-Hill Book Company, Inc. New York, 1957.

3. Annapurna Das, Sisir K Das, Microwave Engineering, McGraw-Hill International Edition, Singapore, 2000.

4. C. G. Montgomery , Techniques of Microwave Measurements , McGraw-Hill, New York, 1947.