# Properties of Dividers and Couplers.

Basic Properties of Dividers and Couplers.
For the remainder of this course we’re going to investigate a plethora of microwave devices and circuits – both passive and active. to begin, during the next six lectures we will focus on different types of power combiners, power dividers and directional couplers. Such circuits are ubiquitous and highly useful. applications include:
• Dividing (combining) a transmitter (receiver) signal to many antennas.
• Separating forward and reverse propagating waves (can also use for a sort of matching).
• Signal combining for a mixer.
As a simple example, a two-way power splitter would have the form

where aÎR and 0£a £1. The same device can often be used as a power combiner:

We see that even the simplest divider and combiner circuits are three-port networks. It is common to see dividers and couplers with even more than that. so, before we consider specific examples, it will be beneficial for us to consider some general properties of three- and four-port networks.
Basic Properties of Three-Port Networks
As we’ll show here, it’s not possible to construct a three-port network that is:
1. lossless,
2. reciprocal, and
3. matched at all ports.
This basic property of three-ports limits our expectations for power splitters and combiners. We must design around it. To begin, a three-port network has an S matrix of the form:

If the network is matched at every port, then S11 = S22 = S33 = 0. (It is important to understand that “matched” means t1 , t2 and t 3= 0 when all other ports are terminated in Z0 .)
If the network is reciprocal, then S21 = S12 , S31 = S13 and S32 = S23 . Consequently, for a matched and reciprocal three-port, its S matrix has the form:

Note there are only three different S parameters in this matrix. Lastly, if the network is lossless, then [S] is unitary. Applying we find that
|S12|2+|S13|2 =1 |S12|2+|S23|2 =1 |S13|2+|S23|2 =1
and applying that
S*13S23=0 S*23S12=0 S*12S13=0
From (6)-(8), it can be surmised that at least two of the three S parameters must equal zero. If this is the case, then none of the equations (3), (4) or (5) can be satisfied. [For example, say S13 = 0. Then (6) and (8) are satisfied. For (7) to be satisfied and S23 ¹ 0, we must have S 12= 0. But with S12 and S13 both zero, then (3) cannot be satisfied.] Our conclusion then is that a three-port network cannot be lossless, reciprocal and matched at all ports. Bummer. This finding has wide-ranging ramifications.
However, one can realize such a network if any of these three constraints is loosened. Here are three possibilities:
1. Nonreciprocal three-port. In this case, a lossless three-port that is matched at all ports can be realized. It is called a circulator :

Notice that Sij¹ Sji .
2. Match only two of the three ports. Assume ports 1 and 2 are matched. Then,

3. Lossy network. All ports can be simultaneously matched and the network reciprocal.
Basic Properties of Four-Port Networks
Unlike three-ports, it is possible to make a lossless, matched and reciprocal four-port network. These are called directional couplers.the S matrix o f a reciprocal and matched four-port has the form

Incorporating the fact that the network is lossless puts further constraints on these S parameters, as discussed in the text. As described in Section 7.1 of the text, there are two commonly used realizations of directional couplers:
1. The Symmetrical Coupler. The S matrix for this device is

Where a,bÎR and a2 +b2 =1. It is obvious from the S matrix that the network is reciprocal and matched. It can also be shown that [S] is unitary, which means this fourport is also lossless.
2. The Asymmetrical Coupler. The S matrix for this device is

We can see from this S matrix that the network is matched and reciprocal. It can also be shown that the network is lossless. we will study this coupler later as the 180º Hybrid.
Directional Couplers
We will now take a quick look at the operation of a directional coupler. Common circuit symbols are :

Or

The arrows indicate the assumed directions of time average
power flow. as we saw in (11) and (12), the S matrix of the symmetrical and antisymmetrical directional couplers has the form

where ‘_’ indicates a non-zero value. we can deduce the operation of this network directly from the S matrix, assuming all the ports are matched.
For example, if power enters port 1, it then splits between ports 2 ( S21 ¹ 0) and 3 ( S31 ¹ 0), while no power is delivered to port 4 (S41 = 0). Since S11 = 0, there will be no reflected power from port 1.
Alternatively, if power enters from port 2, it then splits between ports 1 (S12 ¹ 0) and 4 ( S42 ¹ 0), but none to port 3 (S32 = 0). Of course, no directional coupler is ideal and the S matrix above is only approximately realized in practice. The performance of directional couplers is characterized by the following three values. For these definitions, port 1 is assumed the input, ports 2 and 3 the outputs and port 4 is the isolated port.
1. Coupling, C:
C= 10log10 (P1/P3) = 10log10(1/S31|2)dB
or, using :
C=-20log10|b| dB
2. Directivity, D:
D=10log10(P3/P4 = 10log10 {(P3/P1)/(P4/P1)} = 10log10 (|S31|2/|S41|2)dB
or, using:
D=20log10(|b|/S14)dB
If the directional coupler is ideal, then D ---¥.
3. Isolation, I:
I=10log10(P1/P4 =10log10(1/|S41|2)dB
I=20log10|S14|dB
These three quantities are related by
I = D +C dB