To: File C:\Hagen\430tx\430_WR2100_turnstile.wpd

From: Jon Hagen

Date: 12-16-03 rev Jan. 12, 2004

Subject: Arecibo Turnstile Junctions - Theory, Tuning, Performance

1. 6-Port Turnstile Junction

The turnstile junction was invented and patented by Robert H. Dicke, who described its operation in Principles of Microwave Circuits ( Rad Lab Series, Volume 8). The circular port supports two orthogonal TE11 modes; these modes are indicated on figure below as "Port 5 and "Port 6". The arrows indicate the electric field direction for these modes. The rectangular ports support only the TE10 mode, whose electric field is perpendicular to the broad wall of the guide. If one begins with an empty turnstile, i.e. only the rectangular and circular waveguide structure, it will be found that power applied to a side port will appear at the other side ports and also at Port 5 or Port 6. From symmetry, power applied to Port 1 cannot exit Port 6 and power applied to Port 2 cannot exit Port 5. (The exiting E-field would have no way to decide whether to point with or against the arrow). One can "tune" the device by inserting a structure of posts, concentric with the circular guide, or, in general
with any other structure that, like the side ports, has 4-fold rotational symmetry about the axis of the circular waveguide. Note that the circular guide could also be a square waveguide - or any other waveguide that has four-fold rotational symmetry. With an appropriate tuning structure, the junction will have the following properties:

a. All ports will be matched, i.e. none of the power incident on any port will be reflected, providing the other ports are connected to non- reflecting loads.

b. Power incident on a side port will split three ways: 25% will flow to each adjacent side port and 50% will flow to either Port 5, or Port 6, as explained above. The opposite side port is isolated: no power flows to it.

2. Turnstile Radar Diplexer

When two opposite side ports are properly blocked off (shorted), the turnstile becomes a radar diplexer, i.e. a 4-port device that steers transmitter power into a single antenna and steers the echo received by the antenna into a receiver. To see this, suppose we have built and tuned a turnstile junction. We now install waveguide shorts on two opposite sidearms and we make the electrical length of these shorts differ by 90 degrees. Now we apply power to the transmitter port. Half this power proceeds out Port 5, as stated above. One quarter of the power goes to each shorted side port and is then reflected back into the turnstile. The reflections, as they reenter the turnstile, differ in phase by 180 degrees, due to the round trip through the 90-degree path difference. Thus the reflected waves cancel at both the transmitter port and at the receiver port (maintaining the isolation of the receiver). But each reflection also sends half of its power into Port 6. These voltages are in phase, so the total power arriving at Port 6 is equal to half the transmitter power, just as at Port 5.

If the shorts are moved by equal amounts, either inward or outward, the cancellation of the reflected waves at the transmitter port and receiver port is maintained. However, this kind of simultaneous short adjustment changes the relative phase of the signals leaving through Ports 5 and 6. If the relative phase is set to be +/- 90 degrees, the mutually perpendicular E fields from Ports 5 and 6 combine to become circular polarization. Suppose the short positions are set so that the circular waveguide sends right-hand circular polarization (RHC) to the feed horn. When the signal bounces against a target, which could be an airplane, the ionosphere, an asteroid, etc, the reflected wave will mostly have the opposite sense of circular polarization. As a result, the received signal entering the circular waveguide will have LHC polarization. Symmetry and reciprocity considerations show that this signal will be steered to the turnstile's receiver port. Used this way, the turnstile is an ideal diplexer; all the transmitter power is steered to the antenna and all the received power is steered to the receiver. (In practice, there is always some stray coupling of transmitter power to the receiver port, which necessitates the use of a PIN diode limiter to protect against receiver burnout).

3. Turnstile Orthomode Transducer

The turnstile diplexer described above can be used as an OMT to make a dual-polarization receiving system. One receiver picks up the RHC components of the input signals while the other receiver picks up the LHC components. If the shorts are moved in or out together, preserving the 90 degree difference, the two channels change from RHC and LHC to orthogonal elliptical polarizations to crossed linear polarizations, etc. Thus a turnstile OMT can be set for any desired pair of orthogonal polarizations.

4. Bandwidth - Turnstile OMT as a directional coupler

It is well known (Rad Lab. Vol. 8) that nearly any arbitrary lossless junction of four transmission lines will be a directional coupler; with appropriate matches on the ports, the junction will be a coupler with perfect directionality. A turnstile OMT is such a junction, where two of the 'transmission lines' are the two modes in the circular waveguide. At the frequency where the device is tuned, it is an 'infinite dB' coupler, since all the power entering one of circular guide ports will exit from one of the rectangular ports. But at other frequencies, some of the power entering one of the circular guide ports will get into the other rectangular port. Far enough from the center frequency, the power split can reach 50/50. Nevertheless, because of the properties of directional couplers, there will be, for any frequency, a set of orthogonal polarizations, each of which will excite only one of the rectangular ports.

Thus information is not lost. Polarimetry can be done by taking into account the appropriate orthogonal polarization ellipses at each frequency. Radiometry of unpolarized sources still enjoys the advantage of having two statistically independent signals, from which to estimate the power. The degradation, away from the center frequency, is from mismatch. Since the receivers are designed to have the lowest noise figures when connected to a reflectionless sources, TSYS increases as the OMT outputs present increasing amounts of reflection. Also, far from the center frequency, mismatch will eventually render the polarizations non-orthogonal.

5. Procedures for tuning the center posts

It can be shown (Appendix 1) that a turnstile can be tuned as follows:

Procedure 1. With the circular port and two opposite side ports terminated by reflectionless loads, adjust the post structure to zero the reflection from one of the remaining side ports while simultaneously zeroing the coupling from that port to the opposite side port.

Procedure 2. Same as Procedure 1, except that shorts replace the side port loads. These shorts can have arbitrary positions, but their lengths must differ by 1/4 of the guide wavelength.

Procedure 3. Same as Procedure 1 or 2 except that the load on the circular guide is replaced by a feed horn looking into space or at a reflector dish. The feed horn need not present a perfect

match to the circular waveguide; as long as it has 4-fold axial symmetry, it can be considered as an integral part of the turnstile.

6. Procedures for positioning the shorts

For this adjustment, it is necessary to know the value of the phase shift from a rectangular port to an adjacent rectangular port. This can be measured with a network analyzer while the other side port and the circular waveguide are terminated in reflectionless loads. Alternatively, if the turnstile has been analyzed with a program

such as HFSS (High Frequency Structure Simulator - finite element electromagnetics) , this phase shift will have been calculated. Knowing this phase shift, one simply adds enough round trip delay to the short and back to produce a total phase shift that is an odd multiple of 90 degrees. The shorter short is made just long enough to get this total phase shift. The longer short is then made exactly 90 degrees longer (180 degrees round trip) . Both shorts could be made longer than these values by a multiple of 90 degrees, but this would narrow the useful bandwidth of the device and make it larger.

When the phase shift has not been measured or calculated, the shorts must be positioned experimentally. Only one parameter must be varied, since one short will always be positioned 90 degrees farther out than the other. With the feed horn attached and power entering the transmitter port, adjust the shorts until the desired polarization is observed to exit the circular guide. A linearly polarized probe antenna can be used; as it is rotated in front of the feed horn, the variation in the detected power indicates the ellipticity of the polarization. If circular polarization is desired, one could put a metal cap over the circular waveguide and then simply adjust the shorts for a perfect input match at either of the two side ports.

Appendix 1. Post Tuning Procedure Analysis

The tuning procedures given in Section 4 can be justified by examining the S-Matrix of the 6-port turnstile. Using the port number assignments shown in Figure 1, and taking into account reciprocity and the turnstile's symmetry, the S- Matrix for an arbitrary, i.e. yet untuned, turnstile must have the form

where is the reflection coefficient at the four side ports, is the transmission coefficient between adjacent side ports, is the transmission from any side port to the opposite side port, is the transmission coefficient from a side port to one of the ports (polarizations) in the circular waveguide, and is the reflection coefficient at the ports in the circular waveguide.

Assume now that the tuning procedure has been carried out, i.e. that and have been made equal to zero. The S matrix now becomes

We assume the turnstile is lossless, i.e. made from a perfect conductor. When a device is lossless, sum of the squares of the magnitudes of the elements in any row of the S-matrix must be unity. Likewise, the dot product of any row with the complex conjugate of any other row must be zero. These conditions follow from the conservation of energy.

Applying the second of these relations to columns 1 and 5 we get =0. Since there is coupling from Port 1 to 5, is not zero, so must be zero, i.e. the circular ports are matched.

Next, applying the first of relations to column 5, we find 2| |2 =1 or | |2 =1/2; half of the power incident on a side port goes to the associated port in the circular waveguide.

It follows that 1/4 of the power incident on a side port goes to each adjacent side port. This also follows from applying the first relation to any of the first four columns. Using the same methods, it is easy to show that equivalent post tuning procedures are to adjust the post structure until = = 0 or until = =0. (Making any two of , , and ensures that the third is also zero). Note that a tuned turnstile is completely described by just two real numbers ,arg( ) and arg( ), the phase shift between adjacent side ports and the phase shift from a side port to its associated port in the circular waveguide.

.

Appendix 2. Carriage House Turnstile

The turnstile OMT in the carriage house is the original, installed around 1962. Its dimensions are shown in Figure A2.1. The four side ports are WR2100 waveguides, 21" wide by 10.5" high. The circular waveguide has an inside diameter of 18.5". The post structure consists of three concentric cylinders of diameters 12", 8.500", and 2.000" whose nominal heights (all measured from the floor) are 0.250", 3.812" and 13.812". The heights of the two inner cylinders are adjustable.

For the bare turnstile without the side shorts, at 430 MHz, the S-matrix elements, as estimated with HFSS (model 'oldur430') are:

=.043 @ 28.4 deg (ideally 0)

= .500 @ 139 deg (ideally .5 @ some angle)

= .017 @ 115 deg (ideally 0)

=.706 @ 145 deg (ideally .707 @ some angle)

= .045 @ 106 (ideally 0)

(See Table A2.1 for other frequencies).

The performance of this turnstile OMT, as calculated by HFSS (model 'fulltst1"), is shown in

Table A2.1. Note the change in port number assignments; for the OMT (a four-four device), the receiver port is Port 2, the transmitter (or second receiver) port 1, and the output ports (orthogonal TE11 modes in the circular waveguide) are Ports 3 and 4.

Frequency	S[1,1]_dB	S[1,1]_Phs	S[1,2]_Mag	S[1,2]_Phs	S[1,3]_dB	S[1,3]_Phs
4.00E+08	-17.9	108.0	0.500	171.7	-16.8	-75.6
4.10E+08	-21.6	87.5	0.500	160.7	-22.2	-92.0
4.20E+08	-24.8	61.8	0.500	149.7	-31.0	-118.0
4.30E+08	-26.8	29.9	0.499	138.8	-34.1	101.6
4.40E+08	-27.1	-2.0	0.498	127.8	-24.8	68.7
4.50E+08	-25.9	-26.2	0.497	116.6	-20.4	52.4
4.60E+08	-24.3	-42.3	0.494	105.0	-17.5	38.2
Frequency	S[1,5]_Mag	S[1,5]_Phs	S[5,5]_Mag	S[5,5]_Phs
4.00E+08	0.680	13.5	0.272	100.9
4.10E+08	0.698	-4.3	0.160	83.6
4.20E+08	0.704	-20.3	0.085	77.6
4.30E+08	0.706	-35.3	0.043	105.9
4.40E+08	0.705	-49.6	0.060	148.3
4.50E+08	0.703	-63.5	0.100	150.7
4.60E+08	0.700	-77.1	0.139	141.7

Table A2.1 HFSS Analysis Results of Carriage House Turnstile (OMT w/o side shorts, file

'oldur430". Port Assignments are shown in Figure 1. Note that the side port return loss (S11) and the cross port isolation (S13) are expressed in dB. Matrix elements that are nominally zero, e.g. S16, are not shown. Their HFSS values are typically around .001.

Turnstile to OMT

The necessary shorts to convert the turnstile into an OMT with dual circular polarizations can be found as follows: Since the phase shift from one rectangular port to an adjacent rectangular is 138.8 degrees, a phase delay of 360-138.8 =221.2 degrees, the short short's round trip must provide another 48.8 degrees to get a total of 270. This 48.8 degrees corresponds to a physical length of guide x .5 x 48.8/360 = .5 x 48.8/360 x 36.265" = 2.46". The other short is longer by /4. Note: each short also provides another 180 of phase shift. When Port 1 is excited, the wave in the circular waveguide has Left Hand circular polarization. Port assignments are shown below in Figure A2.2.

The performance of the OMT is show below in Table A2.2.

Frequency	S[1,1]_dB	S[1,2]_dB	S[1,2]_Phs	S[1,3]_Mag	S[1,3]_Phs	S[1,4]_Mag	S[1,4]_Phs	LHC/RHC (dB)
4.00E+08	-10.9	-13.6	-5.6	0.680	12.3	0.642	154.9	6.1
4.05E+08	-13.5	-14.6	-21.3	0.691	3.0	0.666	137.3	7.8
4.10E+08	-16.4	-16.1	-37.0	0.698	-5.6	0.682	120.3	9.8
4.15E+08	-19.7	-18.4	-53.6	0.702	-13.8	0.694	103.8	12.2
4.20E+08	-23.8	-21.9	-72.7	0.705	-21.6	0.702	87.6	15.4
4.25E+08	-29.2	-27.9	-103.4	0.706	-29.3	0.706	71.6	20.4
4.30E+08	-32.0	-32.3	160.4	0.707	-36.7	0.706	55.8	33.1
4.35E+08	-28.2	-24.7	105.1	0.707	-44.0	0.703	40.3	26.0
4.40E+08	-24.8	-20.2	83.4	0.707	-51.2	0.698	25.0	18.3
4.45E+08	-22.4	-17.4	67.6	0.706	-58.3	0.691	9.9	14.3
4.50E+08	-20.6	-15.5	54.1	0.705	-65.3	0.683	-5.0	11.5
4.55E+08	-19.2	-14.1	41.9	0.703	-72.2	0.674	-19.7	9.4
4.60E+08	-18.1	-13.0	30.5	0.701	-79.2	0.665	-34.2	7.6

Table A2.2. Ports 1 and 2 are, respectively, the transmitter, receiver, and antenna ports, as shown in Figure A2.1. . HFSS calculations on the model 'fulltsl" produced the following results: At the center frequency, 430 MHz, the input return loss at the transmitter is 37.2dB. The isolation between the transmitter and receiver ports is 35.1 dB. The amplitudes of the two linearly polarized waves in the output waveguide (antenna port) are virtually equal (.706 and .708). Their phase difference is 124 deg. - 33.7 deg. = 90.3 degrees, almost exactly the 90 degrees needed for pure LHC polarization. The calculated component of RHC power is 51dB below the LHC power. Data for other frequencies is shown in Table A2.2.

Appendix 3. Dome 430MHz Turnstile OMT

To save space, the turnstile in the dome has side ports that are half-height WR2100, i.e. 21' x 5.25", and the longer side short is folded. The inside diameter of the circular waveguide is 20". A corona bal1 with a l .5" diameter is placed at the end of the 1" diameter center post. HFSS-calculated performance of the bare turnstile (no side shorts) is shown in Table A3.1. The as-built OMT originally performed best at 435 MHz so a metal 'puck', 4" diameter x 3/4" high, was put at the end of the short short to bring the frequency down to 430 MHz. The design is shown below in Figure A3.1. The OMT performance, as calculated by HFSS on model "tnst01pk" is shown in Figure A3.2.

Frequency	S[1,1]_dB	S[1,1]_Phs	S[1,2]_Mag	S[1,2]_Phs	S[1,3]_dB	S[1,3]_Phs
4.00E+08	-18.1	-132.0	0.483	173.1	-15.5	-71.4
4.05E+08	-18.5	-134.0	0.486	166.8	-16.8	-78.0
4.10E+08	-19.1	-136.2	0.489	160.4	-18.5	-84.6
4.15E+08	-19.8	-138.8	0.492	153.8	-20.6	-91.1
4.20E+08	-20.9	-141.6	0.495	147.0	-23.6	-96.9
4.25E+08	-22.5	-144.2	0.497	139.7	-28.8	-99.6
4.30E+08	-25.3	-145.5	0.499	131.9	-42.5	-37.3
4.35E+08	-31.4	-136.4	0.500	123.4	-28.0	39.6
4.40E+08	-33.3	-31.1	0.498	113.7	-21.1	34.9
4.45E+08	-22.5	-12.8	0.489	102.4	-16.7	24.7
4.50E+08	-16.3	-19.3	0.467	88.9	-13.2	11.2
4.55E+08	-11.8	-32.1	0.420	72.4	-10.4	-5.8
4.60E+08	-8.5	-49.4	0.335	52.6	-8.3	-26.2
Frequency	S[1,5]_Mag	S[1,5]_Phs	S[5,5]_Mag	S[5,5]_Phs
4.00E+08	0.699	-47.0	0.153	-68.3
4.05E+08	0.702	-56.5	0.126	-86.8
4.10E+08	0.704	-65.7	0.101	-106.9
4.15E+08	0.705	-74.6	0.080	-129.9
4.20E+08	0.706	-83.4	0.064	-157.9
4.25E+08	0.706	-91.9	0.056	168.1
4.30E+08	0.706	-100.4	0.057	132.5
4.35E+08	0.706	-108.6	0.067	101.9
4.40E+08	0.705	-116.8	0.081	77.7
4.45E+08	0.704	-124.8	0.098	58.1
4.50E+08	0.703	-132.8	0.116	41.5
4.55E+08	0.701	-140.6	0.134	26.7
4.60E+08	0.699	-148.4	0.153	13.1

Table A3.1 HFSS Analysis Results of Dome Turnstile (OMT w/o side shorts, file

'tnst000"). Port Assignments are shown in Figure 1. Note that the side port return loss (S11) and the cross port isolation (S13) are expressed in dB. Matrix elements that are nominally zero, e.g. S16, are not shown. Their HFSS values are typically around .001.

The calculated performance of the dome OMT (Turnstile with side shorts) is shown below in Table A3.2. Note that the performance hardly differs from that of the carriage house (full height) turnstile. At the center frequency, the polarizations are essentially pure circular. At 460 MHz they have become pure linear.

Frequency	S[1,1]_dB	S[1,2]_dB	S[1,3]_Mag	S[1,3]_Phs	S[1,4]_Mag	S[1,4]_Phs	LHC/RHC (dB)
4.00E+08	-24.1	-18.5	0.700	-91.3	0.702	54.7	5.5
4.05E+08	-33.5	-18.1	0.702	-101.8	0.701	35.1	7.2
4.10E+08	-38.6	-18.6	0.704	-112.1	0.700	16.0	9.2
4.15E+08	-29.7	-20.0	0.706	-122.1	0.701	-3.0	11.7
4.20E+08	-27.4	-22.6	0.707	-132.0	0.702	-22.1	15.1
4.25E+08	-27.7	-28.1	0.707	-141.6	0.705	-41.5	21.1
4.30E+08	-30.5	-30.2	0.707	-151.1	0.705	-61.7	45.3
4.35E+08	-28.9	-21.4	0.707	-160.4	0.701	-82.8	19.3
4.40E+08	-21.5	-16.1	0.707	-169.6	0.685	-105.3	12.8
4.45E+08	-16.0	-12.4	0.706	-178.7	0.648	-129.4	8.6
4.50E+08	-12.1	-9.9	0.704	172.3	0.582	-154.9	5.1
4.55E+08	-9.3	-8.1	0.703	163.4	0.483	178.4	2.1
4.60E+08	-7.3	-7.1	0.701	154.6	0.356	151.2	-0.4

Table A3.2 HFSS Analysis Results of Dome Turnstile OMT, file

'tnst01pk". Port Assignments are shown in Figure A2.2. Note that the side port return loss (S11) and the cross port isolation (S13) are expressed in dB. Matrix elements that are nominally zero, e.g. S16, are not shown. Their HFSS values are typically around .001.

Appendix 4. S-band Turnstile OMT

The S-band turnstile, without side shorts, is shown

in Figure A4.1. The rectangular waveguides are

WR430, whose width and height are, respectively,

4.300" and 2.150". The distance between opposite

rectangular ports is 6.300 inches. Dimensions of the

probe structure (and shorts) are shown in Figure A4.2. The results of the HFSS analysis on the bare turnstile are shown below in Table A4.1.

Frequency	S[1,1]_dB	S[1,1]_Phs	S[1,2]_Mag	S[1,2]_Phs	S[1,3]_dB	S[1,3]_Phs
2.32E+09	-24.4	6.4	0.496	96.9	-23.0	6.7
2.34E+09	-21.3	-4.4	0.492	90.4	-20.7	4.4
2.36E+09	-18.9	-13.5	0.487	83.6	-18.7	0.4
2.38E+09	-16.8	-21.9	0.480	76.6	-16.9	-4.9
2.40E+09	-15.0	-30.2	0.470	69.2	-15.4	-11.1
2.42E+09	-13.4	-38.5	0.458	61.4	-14.0	-18.3
2.44E+09	-11.8	-47.0	0.439	53.0	-12.7	-26.0
Frequency	S[1,5]_Mag	S[1,5]_Phs	S[5,5]_Mag	S[5,5]_Phs
2.32E+09	0.707	117.6	0.008	-133.3
2.34E+09	0.707	107.8	0.015	145.1
2.36E+09	0.707	98.1	0.028	114.8
2.38E+09	0.706	88.5	0.043	95.4
2.40E+09	0.706	78.9	0.059	79.0
2.42E+09	0.705	69.4	0.076	63.7
2.44E+09	0.704	59.9	0.093	49.6

Table A4.1 HFSS Analysis Results of Carriage House Turnstile (OMT w/o side shorts, file

'tnst23mt". Port Assignments are shown in Figure 1. Note that the side port return loss (S11) and the cross port isolation (S13) are expressed in dB. Matrix elements that are nominally zero, e.g. S16, are not shown. Their HFSS values are typically around .0015.

The full 2380 OMT (Turnstile with side shorts) is shown below in Figure A4.2

HFSS analysis results of the OMT (turnstile with shorts) are shown below in Table A4.2.

Frequency	S[1,1]_dB	S[1,2]_dB	S[1,3]_Mag	S[1,3]_Phs	S[1,4]_Mag	S[1,4]_Phs	LHC/RHC (dB)
2.32E+09	-21.8	-21.424	0.707	116.0	0.697	-139.8	18.1
2.34E+09	-20.7	-21.294	0.707	106.1	0.696	-157.1	24.5
2.36E+09	-19.2	-19.986	0.707	96.3	0.691	-174.8	36.6
2.38E+09	-17.2	-17.852	0.707	86.6	0.682	167.1	21.4
2.40E+09	-15.2	-15.533	0.708	76.6	0.664	148.4	15.7
2.42E+09	-12.9	-13.574	0.704	67.4	0.640	129.8	12.0
2.44E+09	-11.1	-11.848	0.703	57.8	0.603	110.5	9.2

Table A4.2 HFSS Analysis of S-band Turnstile OMT (file"trnst23a)

Note that the predicted performance of the S-Band OMT is mediocre; the return loss and isolation are only about -17dB at 2380 MHz. Performance improves below 2380 MHz. Perhaps there are errors in the measured dimensions. The unit should be tested in the lab.

Appendix 5. Sense of Circular Polarization

All the turnstiles described above will output LHC polarization when the shorts are installed as shown below in figure A5.1. The shorts were assumed to be installed this way in analysis of each turnstile. (If the shorts are interchanged the turnstile will output RHC). The polarization leaving the AO dish will, however, be changed from LHC to RHC because the wave is further reflected an odd number of times. (A wave leaving the line feed is reflected once, from the main dish. A wave leaving the S-band horn is reflected from the Gregorian tertiary and secondary reflectors, and then from the main dish.

Definition of LHC polarization: The E vector, as seen when the wave is approaching us rotates in the clockwise direction. By (IRE, IEEE) definition this is left-circular polarization. Radio astronomers use this definition.

The ratios of LHC power to RHC power shown in Tables A2.2, A3.2, and A4.2 are calculated as follows: Referring to Figure A2.2, let the arrow defining Port 3 be the x axis and the arrow defining Port 4 be the y axis. The circular guide points in the Z directioin.S13 and S14 are therefore the X and Y electric fields. The LHC E-field can be written as EL(x + jy), where EL is a complex amplitude and x and y are unit vectors. Likewise, the RHC E-field can be written as ER(x - jy). We can therefore write down the X and Y fields as

Solving for EL and ER, we have

Finally, the ratio of RHC to LHC power is just |EL/ER|2.

Appendix 6. OMT Performance calculated from Turnstile Performance

The OMT results presented in Tables A2.2, A3.2, and A4.2 were calculated by HFSS models of the OMTs, i.e turnstiles with side shorts. But if one knows the S-parameters of the turnstile, the OMT performance can be calculated by combining these parameters with the extremely simple

reflection coefficients (S11) of the shorts. Let X be a 6-element vector representing the waves incident at each of the 6 ports of the turnstile and let Y be a 6-element vector representing the resulting output waves. In matrix notation, Y=SX. Now we attach shorted waveguide sections to Ports 2 and 4. These shorts have reflection coefficients -e-jph1 and -e-jph2 , where ph1 and ph2 are the electrical lengths in radians and the minus signs account for the phase reversal at the shorts. Because of the shorts on Ports 2 and 4, we have that X2 = -e-jph1Y2 and X4 = -e-jph2 Y4. The excitation vector is therefore has 4 elements corresponding to the still open ports and 2 elements proportional to Y.

We can write this as S(X + Gamma Y) = Y, where Gamma is a diagonal matrix whose second and fourth elements are -e-jph1 and -e-jph2. Solving for Y we find Y = (1- S Gamma)-1S X. The S-
matrix of the OMT is therefore given by SOMT = (1- S Gamma)-1S. This is still a 6x6 matrix, but we simply discard the 2nd and 4th rows and columns to obtain the desired 4 x 4 matrix.

The next page is a Mathcad worksheet showing this applied to the carriage house turnstile at 430 MHz.

Since the turnstile is tuned for this particular frequency, all the non-zero S matrix elements are either

S12 or +/- S15, whose values, taken from Table A2.1 are .499 @ 138.8 deg. and .706 @ -35.3 degrees. The calculated values of S13 and S14 for the OMT are .706@-35.72 deg and

.705 @ 54.715 deg. These are in good agreement with the HFSS calculations from Table A2.2, namely, .707@ -36.7 deg and .706 @ 55.8 degrees. Equally good agreement should be produced for other frequencies.

Appendix 7 - Empty Turnstile

The Carriage House turnstile of Figure A2.1 was analysed empty, i.e. without the post structure.

Results are shown below.

Frequency	S[1,1]_dB	S[1,1]_Phs	S[1,2]	S[1,2]_Phs	S[1,3]_db	S[1,3]_Phs	S[1,5]	S[1,5]_Phs
430MHz	-18.96	155.83	4.67E-01	106.73	-6.90	7.32	5.90E-01	-92.70
S[5,5]_dB	S[5,5]_Phs
5.51E-01	173.42

Note that the input match at the side ports (S11) is fairly good. The amout of power steered to adjacent side ports is still about 25% each. But 20.4% of the power exits the no-longer isolated opposite port.

Appendix 8 - Carriage House Alternate Turnstile

On Jan. 20, 2004, the carriage house turnstile was inspected and tuned. It was found that this turnstile

is mechanically somewhat different from the spare turnstile described above aas the carriage house turnstile. A drawing of the turnstile actually installed in the carriage house is shown below in Figure A8.1.