Magic Tee Radar Applications

Magic Tee Power Combining Using Two Transmitters

The following questions are different versions of the same practical application of a Magic Tee. In each case, the requirement is to use two identical microwave transmitters so that their available power can be combined and supplied to a single antenna, while the transmitters do not load or interfere with each other. The required arrangement is obtained by connecting the two transmitters to the two auxiliary arms of the Magic Tee, namely Port 3 and Port 4. Because the two arms have different phase characteristics, the waves produced by the two transmitters cancel at one collinear port and add at the other. The port at which the waves add constructively is connected to the antenna, allowing the combined transmitter power to be delivered to the antenna.

Solution: Using a Magic Tee to Combine Two Transmitters

A suitable solution to all of the above problems is to use a Magic Tee as a power combiner. The Magic Tee allows the outputs of two identical microwave transmitters to be combined at one collinear port while maintaining isolation between the two transmitter ports. For this purpose, the two transmitters are connected to Port 3 and Port 4 of the Magic Tee respectively. The two remaining collinear ports, Port 1 and Port 2, are then used to obtain the required cancellation and combined output. Port 1 becomes the port at which the signals from the two transmitters cancel, while Port 2 becomes the port at which the signals add constructively. The antenna is therefore connected to Port 2 so that the combined microwave power is delivered to it.

The reason this arrangement works is based on the different phase relationships associated with excitation of the two auxiliary arms of a Magic Tee. When a microwave signal is applied to the E-arm, which is represented by one of the auxiliary ports, equal-amplitude signals appear at the two collinear arms with a \(180^\circ\) phase difference. When a microwave signal is applied to the H-arm, equal-amplitude signals appear at the two collinear arms with the same phase. By connecting one transmitter to the E-arm and the other transmitter to the H-arm, the two transmitters produce exactly the phase relationships required for cancellation at one collinear port and addition at the other.

1. Connecting the Two Transmitters to Port 3 and Port 4

The first step in solving the problem is to connect the two identical transmitters to the two auxiliary ports of the Magic Tee. Let Transmitter 1 be connected to Port 3 and Transmitter 2 be connected to Port 4. Ports 1 and 2 are the two collinear ports of the Magic Tee. Port 2 is selected as the antenna port because the signals from the two transmitters will be in phase at this port and therefore combine constructively. Port 1 is the cancellation port because the contributions from the two transmitters arrive there with opposite phase.

Thus, the basic arrangement can be represented as

\[ \text{Transmitter 1}\rightarrow Port\ 3 \]

\[ \text{Transmitter 2}\rightarrow Port\ 4 \]

\[ Port\ 1\rightarrow \text{Cancellation} \]

\[ Port\ 2\rightarrow \text{Antenna} \]

This connection is important because the two transmitters must not simply be connected together at the same microwave junction. A direct connection could cause one transmitter to interact with the output impedance of the other transmitter, resulting in undesirable loading, reflections, frequency instability, or interference between the two sources. The Magic Tee avoids this problem by using its hybrid properties to keep the two transmitter ports isolated while combining their contributions at the selected output port.

The purpose of connecting the transmitters to Port 3 and Port 4 is therefore not merely to provide two input paths. The two ports have complementary phase characteristics. One excitation generates an opposite-phase pair at the collinear ports, while the other generates an in-phase pair. These two behaviors are deliberately combined so that the resulting waves have different effects at Port 1 and Port 2.

2. Wave Produced by Transmitter 1 Connected to Port 3

Consider first the operation of Transmitter 1, which is connected to Port 3. When Transmitter 1 supplies a microwave signal to Port 3, the Magic Tee distributes this signal between the two collinear ports, Port 1 and Port 2. The two resulting waves have equal magnitude but are opposite in phase. Therefore, if the wave produced at Port 1 is represented by \(A\), the corresponding wave at Port 2 can be represented by \(-A\).

The signal relationship produced by Transmitter 1 can therefore be written as

\[ b_{1T_1}=A \]

and

\[ b_{2T_1}=-A \]

where \(A\) represents the corresponding wave amplitude after the power division through the Magic Tee. The negative sign does not mean that the power is negative. It represents a \(180^\circ\) phase reversal between the two waves. The two waves therefore have equal power but opposite phase.

This is the characteristic difference-mode behavior of the E-arm. A signal entering the E-arm divides equally between the two collinear arms, but the fields at the two collinear ports have opposite phase. This phase reversal is the key property that will later cause cancellation at one port when the second transmitter is introduced.

3. Wave Produced by Transmitter 2 Connected to Port 4

Now consider Transmitter 2, which is connected to Port 4. The excitation at Port 4 produces equal-amplitude waves at Port 1 and Port 2, but in this case the two waves are in phase. If the wave produced at Port 1 is represented by \(A\), the wave produced at Port 2 is also represented by \(A\). Therefore, the signal contributions produced by Transmitter 2 can be written as

\[ b_{1T_2}=A \]

and

\[ b_{2T_2}=A \]

Here, both waves have the same magnitude and the same phase. This is the sum-mode behavior associated with the H-arm of the Magic Tee. A signal entering the H-arm is divided equally between the two collinear arms, and the resulting waves have the same phase.

The importance of this second excitation becomes clear when its waves are combined with those produced by Transmitter 1. Transmitter 1 produces opposite-phase contributions at Ports 1 and 2, whereas Transmitter 2 produces in-phase contributions. Consequently, the two transmitter signals do not simply add equally at both output ports. Instead, their phase relationships cause cancellation at one port and constructive addition at the other.

4. Cancellation of Waves at Port 1

At Port 1, the wave produced by Transmitter 1 and the wave produced by Transmitter 2 have opposite phase. From the previous results, the contribution from Transmitter 1 is \(A\), while the contribution from Transmitter 2 is also of equal magnitude but with the phase relationship required for cancellation at this port. The total wave at Port 1 is therefore obtained by superposition of the two contributions.

The total output wave at Port 1 can be written as

\[ b_1=b_{1T_1}+b_{1T_2} \]

With the required opposite-phase relationship,

\[ b_1=A-A \]

Therefore,

\[ \boxed{b_1=0} \]

Thus, the waves arriving at Port 1 cancel completely under ideal conditions. The cancellation occurs because the two transmitter signals have equal magnitude and \(180^\circ\) relative phase at this port. Since the resultant wave is zero, no net signal is delivered through Port 1 in the ideal case. Port 1 is therefore referred to as the cancellation port for this particular configuration.

This cancellation is also an important part of the isolation mechanism. The transmitter signals are arranged through the hybrid properties of the Magic Tee so that their energy does not simply propagate back into the other transmitter path. Instead, the fields interfere according to their phase relationship, producing a null at the selected collinear port. This allows the Magic Tee to combine the transmitter outputs without directly connecting the two transmitter sources together.

5. Addition of Waves at Port 2

At Port 2, the situation is different. The wave contributed by Transmitter 1 and the wave contributed by Transmitter 2 arrive with the same phase. Therefore, instead of cancelling, the two waves add constructively. The total wave at Port 2 is given by

\[ b_2=b_{2T_1}+b_{2T_2} \]

For the selected phase relationship, the two contributions have the same sign and equal magnitude, so

\[ b_2=A+A \]

Hence,

\[ \boxed{b_2=2A} \]

The waves therefore add constructively at Port 2. This is the reason Port 2 is selected as the antenna port. The antenna receives the combined output produced by the two transmitters rather than the output of only one transmitter. The Magic Tee has consequently converted the two separate transmitter outputs into a combined microwave output at a single port.

6. Why Double Output Power Is Obtained at the Antenna Port

The important point in the examination problem is that the antenna requires twice the power available from either individual transmitter. The Magic Tee provides the required power-combining action by allowing the two equal transmitter powers to be directed toward the same antenna port while the unwanted component is cancelled at the other collinear port. If each transmitter is capable of supplying an available power \(P\), then the total available input power from the two transmitters is

\[ P_{\mathrm{total}}=P+P=2P \]

Under ideal lossless conditions, the combined power delivered to the antenna can therefore reach the required value of

\[ \boxed{P_{\mathrm{antenna}}=2P} \]

Thus, if either transmitter alone can provide \(P\) watts but the radar antenna requires \(2P\) watts, two identical transmitters can be combined using the Magic Tee to meet the antenna power requirement. The important qualification is that the power combination is achieved through the hybrid phase relationships of the Magic Tee, not by simply joining two transmitter outputs together.

The apparent doubling of the wave amplitude in the simplified representation should also be interpreted carefully. In a physical Magic Tee, the \(1/\sqrt{2}\) factors associated with equal power division ensure that the power relationships remain consistent with conservation of energy. The useful result is that each transmitter contributes its available power to the combining process, and under the ideal matched and lossless conditions the two transmitter powers can be delivered together at the antenna port.

7. Why the Two Transmitters Do Not Load Each Other

A major requirement of the problem is that the two transmitters should not load each other. This means that the output of one transmitter should not behave as an unwanted load for the other transmitter. Directly connecting two microwave transmitters together would generally not provide the required isolation because each transmitter would interact with the impedance and signal generated by the other transmitter. Such interaction could cause reflections and unwanted coupling between the sources.

The Magic Tee avoids this direct interaction by using its hybrid properties. The two transmitters are connected to different auxiliary ports, Port 3 and Port 4, rather than directly to the same output junction. Their signals are transformed into specific phase relationships at the collinear ports. One output port experiences cancellation, while the other receives the combined signal. Therefore, the transmitters can operate together as independent sources while their powers are combined at the antenna port.

The isolation can also be understood from the ideal S-matrix. The appropriate scattering coefficient between the two auxiliary ports is zero, meaning that an excitation applied at one transmitter port does not directly appear at the other transmitter port. In the ideal Magic Tee,

\[ S_{34}=S_{43}=0 \]

This isolation is fundamental to the power-combining application. It allows the two transmitters to be coupled to a common antenna through the Magic Tee without directly loading one another.

8. Final Operation of the Two-Transmitter Magic Tee Arrangement

The complete operation can therefore be understood by following the two transmitter signals simultaneously. Transmitter 1 is connected to Port 3 and produces equal-amplitude waves at Ports 1 and 2 with opposite phase. Transmitter 2 is connected to Port 4 and produces equal-amplitude waves at Ports 1 and 2 with the same phase. When the contributions from both transmitters are superimposed, the phase relationship is such that the waves cancel at Port 1. At Port 2, the corresponding waves are in phase and therefore add constructively. Port 2 is connected to the antenna, so the combined transmitter power is delivered to the antenna.

The complete signal-flow principle can therefore be summarized as

\[ \text{Transmitter 1}\rightarrow Port\ 3 \rightarrow \begin{cases} Port\ 1:\text{ one phase}\\ Port\ 2:\text{ opposite phase} \end{cases} \]

while

\[ \text{Transmitter 2}\rightarrow Port\ 4 \rightarrow \begin{cases} Port\ 1:\text{ corresponding opposite phase}\\ Port\ 2:\text{ corresponding same phase} \end{cases} \]

Consequently,

\[ \boxed{\text{Port 1: destructive interference}} \]

and

\[ \boxed{\text{Port 2: constructive interference}} \]

Therefore, Port 2 is connected to the antenna, and the two transmitter outputs are combined to provide the required increased antenna input power. This is the central solution to the 2074 Magh and 2073 Bhadra examination problems and represents the same underlying Magic Tee principle tested in the related radar and duplexer question.