1. EKT 441 MICROWAVE COMMUNICATIONS
CHAPTER 5:
POWER DIVIDER & COUPLERS

2. INTRODUCTION
In some cases, power from a single microwave power amplifier may be insufficient to power up a device (e.g a radar transmitter)
In such case, several power amplifiers may be used with their power added using power combiners
Microwave couplers allows construction of balanced amplifiers that feature constant gain over wide bandwidth
Passive 3- and 4-port passive components are used to solve problems like this, which will be the main focus of this chapter.

3. INTRODUCTION Three Port Network S Parameter Matrix
Review of Power Dividers and Couplers
Directional Coupler
Hybrid Couplers
The Lossless T Junction
Junction T Example
The Resistive Divider
Scattering Matrix of the Resistive Divider

4. INTRODUCTION
Power dividers and couplers are passive microwave components used for power division or power combining.
Power division: an input signal is divided by the coupler into two (or more) signal or lesser power.
Power divider types:
equal division (3 dB) type (T-junctions & circulators)
unequal power division type (Resistive power divider)

5. INTRODUCTION Directional coupler – an arbitrary power division.
Hybrid junctions – equal power division.
Hybrid junctions have either:
a 90º (quadrature) or
a 180º (magic-T) phase shift between the output ports.
Some general properties of the 3- and 4-port networks will be first analyzed and discussed.

6. Three Port Network S Parameter
The scattering matrix of an arbitrary three port network has nine independent variables:
If all the components of the network are passive and it does not contain any active component, then its S parameter matrix must be reciprocal. The [S] matrix will also be symmetric.
Usually to avoid power loss, we would like to have a network that is matched at all ports and is lossless.
However, it is impossible to construct a three port lossless reciprocal network that is matched at all ports.

7. Three Port Network S Parameter
If all the three ports are matched the [S] matrix can be written as:
If the three port network is not reciprocal then and its [S] matrix will not be symmetric.

8. Three Port Network S Parameter
The simplest type of power divider is a T-junction, which is a three-port network with two inputs and one output.
The scattering matrix of an arbitrary three-port network has nine independent elements.
[5.1]
If the component is passive and contains no anisotropic materials, then it must be reciprocal and its [S] matrix must be symmetric (Sij = Sji).
Usually, to avoid power loss, we would like to have a junction that is lossless and matched to all ports.

9. Three Port Network S Parameter
If all ports are matched, the Sii = 0, and if the network is reciprocal the scattering matrix of [5.1] reduces to:
[5.2]
[5’]
If the network is lossless, then the energy conversion [5’] requires that the scattering matrix be unitary, which leads to the following conditions:

10. Three Port Network S Parameter
[5.3b]
[5.3c]
[5.3d]
[5.3e]
[5.3f]
[5.3d-f] show that at least two of the three parameters (S12, S13, S23) must be zero.

11. Three Port Network S Parameter
If the three-port network is nonreciprocal, then Sij ≠ Sji, and the condition of the input matching at all ports and energy conservation can be satisfied.
Such device is known as a circulator, and generally relies on an anisotropic material, such as ferrite to achieve nonreciprocal behavior.

12. THREE-PORT NETWORKS (T-JUNCTIONS)
Figure 5.2: the two types of circulators and their [S] matrices. (the phase references for the ports are arbitrary) (a) Clockwise circulation. (b) Counterclockwise circulation.

13. THREE-PORT NETWORKS (T-JUNCTIONS)
Figure 5.3: A reciprocal, lossless three-port network matched at ports 1 and 2

14. Three Port Network S Parameter
Then if the network is lossless, [S] must be unitary, which implies the following:
[5.4a]
[5.4b]
[5.4c]
[5.4d]
[5.4e]
[5.4f]

15. Three Port Network S Parameter
These equations can be satisfied in one or two ways. Either:
[5.5a]
[5.5b] or
[5.5c]
[5.5d]
This result shows that Sij ≠ Sji for i ≠ j, which implies that the device must be nonreciprocal.

16. INTRODUCTION Three Port Network S Parameter Matrix
Review of Power Dividers and Couplers
Directional Coupler
Hybrid Couplers
The Lossless T Junction
Junction T Example
The Resistive Divider
Scattering Matrix of the Resistive Divider

17. Power Dividers and Couplers
In low frequency design, it is common to sum signals by adding their currents or voltages. In high frequency design direct summation is not used. Instead there is a class of circuits used to sum or split power.
Power Splitter
Power Combiner

18. Power Dividers and Couplers
Two basic types of these circuits are: Power Divider and Power Combiner.
Three port networks are shown here, but both dividers and combiners can be multi-port networks. The most common value for  in splitter is –3 dB (when P2 = P3 ). The power ratio in splitter can range up to –40 dB for one path.

19. Power Dividers and Couplers
Power division and combining can be achieved with 3 port networks P1
Divider or
coupler
P2=aP1
Power division
P3=(1-a)P1
P1=P2+P3
Divider or
coupler P2
Power combining P3
a is known as the division ratio

20. Power Dividers and Couplers
An example of 1 to 4 power divider;
a2a1P1
1 to 2 Power
Divider
a1P1
(1-a2)a1P1 P1
1 to 2 Power
Divider
a2(1-a1)P1
1 to 2 Power
Divider
(1-a2)(1-a1)P1
(1-a1)P1

21. INTRODUCTION Three Port Network S Parameter Matrix
Review of Power Dividers and Couplers
Directional Coupler
Hybrid Couplers
The Lossless T Junction
Junction T Example
The Resistive Divider
Scattering Matrix of the Resistive Divider

22. Four Port Networks
The [S] matrix of a reciprocal four-port network matched at all ports has the following form:
[5.6]

23. Four Port Networks
For an ideal coupler, this network will be reciprocal, matched at each port, and will have no power delivered to the isolated port. The [S] matrix becomes

24. Directional Coupler
One specific class of power divider is the directional coupler. This is a four port device that samples the power flowing into port 1 coupled in to port 3 (the coupled port) with the remainder of the power delivered to port 2 (the through port) and no power delivered to the isolated port.
Directional Coupler
Input
Through
Isolated
Coupled
Usually the isolated port is terminated within the coupler casing. In such case, the coupler appears to be a three port device. In ideal case no power is delivered to port 4 (the isolated port).

25. Directional Coupler
Directional Coupler
Input
Through
Isolated
Coupled
Directional couplers are described by three specifications:
Coupling (C) - The ratio of input power to the couple power.
Directivity (D)- The ratio of coupled power to the power at the isolated port.
Isolation (I) – The ratio of input power to power out of the isolated port.

26. Directional Coupler
Figure 5.3: two commonly used symbols for directional couplers and power flow conventions.

27. Directional Coupler
The basic operation of a directional coupler can be illustrated with the aid of Figure 5.3, which shows two commonly used symbols for a directional coupler and the port definition.
A four port device with a property that a wave incident in port 1 couples ports 2 and 3 but not into port 4 as shown below
The structure has four ports input,direct(through), coupled and isolated.
The 3 important parameters describing the performance of coupler are;
C (coupling factor),
D (directivity) and
I (Isolation)

28. Directional Coupler
The following three quantities are generally used to characterize a directional coupler:
1. Coupling (C) – relates the coupled power (P3) to the input power (P1), that is;
If all the ports are terminated in matched loads, the coupling coefficient becomes simply:

29. Directional Coupler
2. Directivity (D) – characterizes the coupler’s ability to direct energy only to the desired port.
When all ports are matched, it is given by;

30. Directional Coupler
3. Isolation (I) – is a measure of how much input power exits the isolated port. It is given by;
When all ports are matched, it is given by;
Coupling and directivity level are also related to the isolation by the following;

31. Directional Coupler
Another important parameter is the main line loss, also known as Insertion Loss (IL). This is the level of loss that is occurred by the coupler between the input (P1) and through (P2) ports.
or for an perfectly matched network;

32. Hybrid Couplers
Hybrid couplers are special cases of directional couplers, where the coupling factor is 3 dB.
There are two types of hybrids.
1) The quadrature hybrid has a 90 degree phase shift between port 2 and 3 when fed from port 1, with the following [S] matrix.
2) The magic-T hybrid or rat-race hybrid has a 180 degree phase shift between port 2 and 3 when fed from port 4, with the following [S] matrix:

33. Example
Hybrid couplers are special cases of directional couplers, where the coupling factor is 3 dB.
There are two types of hybrids.
1) The quadrature hybrid has a 90 degree phase shift between port 2 and 3 when fed from port 1, with the following [S] matrix.
2) The magic-T hybrid or rat-race hybrid has a 180 degree phase shift between port 2 and 3 when fed from port 4, with the following [S] matrix:

34. INTRODUCTION Three Port Network S Parameter Matrix
Review of Power Dividers and Couplers
Directional Coupler
Hybrid Couplers
The Lossless T Junction
Junction T Example
The Resistive Divider
Scattering Matrix of the Resistive Divider

35. T Junction
Figure 5.4: Various T-junction power dividers. (a) E plane waveguide T. (b) H plane waveguide T. (c) Microstrip T-junction.

36. The Lossless T Junction
The simplest type of power divider is the T junction. T junction can be implemented using virtually any type of transmission line. However, the T junction is very simple to implement, it must be treated with care because it does not offer any isolation between its ports.
Matching requirement for the T junction input:

37. The Lossless T Junction
In general, the fringing fields and higher order modes associated with the discontinuity at such a junction, leading to store energy that can be accounted for by a lumped susceptance, B.
In order for divider to be matched to the input line of characteristic impedance, Z0, it must have:
[5.8]
If the transmission lines are assumed to be lossless (or low loss), then the characteristic impedances are real. If assume that B = 0, then [5.8] reduces to:
[5.9]

38. The Lossless T Junction
In order for the input port to be matched, the output lines must be matched (terminated in their characteristic impedance).
The power dividing ratio can be selected by using different values of characteristic impedance for ports 2 and 3.

39. The Lossless T Junction
Input matching requirement:
Therefore, the input to the T junction can be matched through the correct choice of impedances in port 2 and 3. What about matching of port 2 and 3?

40. The Lossless T Junction
Now lets consider matching of port 2.
If port 2 is matched, then 2 = 0 and we will have:
Substitution of yields Z2 = 0.
Which shows that the lossless T junction cannot be matched at all three ports simultaneously. (Problem 7.1 Pozar)

41. Junction T Example
The resistive power divider for an equal power split.
If the T junction contains lossy components then it is possible to match all the three ports. In this case the signal power will be reduced due to loss in the junction.

42. INTRODUCTION Three Port Network S Parameter Matrix
Review of Power Dividers and Couplers
Directional Coupler
Hybrid Couplers
The Lossless T Junction
Junction T Example
The Resistive Divider
Scattering Matrix of the Resistive Divider

43. The Resistive Divider
Assuming that all the lumped-element resistors are terminated in the characteristic impedance Zo, the input impedance looking into any port is:
Since the network is symmetric from all three ports, the output ports are also matched. Therefore, S11=S22=S33=0

44. The Resistive Divider The voltage at the center of the junction is:
The output voltages V2 and V3 are equal to:

45. Scattering Matrix of the Resistive Divider
Since the network is symmetric and reciprocal, S21=S31=S23=1/2.
Thus, the output power is –6 dB below the input power level (lossy).
The power delivered to the input and outputs of the divider are:
Half of the supplied power is dissipated in the resistors.

46. Example – T Junction
A lossless T junction power divider has a source impedance of 50 Ohms. Find the output characteristics impedance so that the input power is divided in a 2:1 ratio. Compute the reflection coefficients seen looking into the output ports.

47. Example – T Junction
If the voltage at the junction is V0, as shown in the figure, the input power to the divider is
Then the output powers are:

48. Example – T Junction
The results yields the characteristic impedance as:
The input impedance to the junction is;
Looking into the 150 Ω output line, we see an impedance of 50||75 = 30 Ω, while at the 75 Ω output line, we see an impedance of 150||50 = 37.5 Ω.

49. Example – T Junction
Thus the reflection coefficient looking into these ports are:

50. Example – T Junction
Including quarterwave transformer into the T junction power divider and compensation
l/4 P2
Zc = 50 ohm
150 ohm
86.6 50 P1
l/4
75 ohm P2
Recommended quarter wavelength
61.24 50

51. THE WILKINSON POWER DIVIDER
Figure 5.6: The Wilkinson power divider. (a) an equal-split Wilkinson power divider in microstrip form. (b) Equivalent transmission line circuit.

52. THE WILKINSON POWER DIVIDER
The Wilkinson power divider is such a network, with the useful property of being lossless when the output ports are matched; that is, only reflected power is dissipated.
The Wilkinson power divider can be made with arbitrary power division, but we will first consider the equal-split (3 dB) case.
This divider is often made in microstrip or stripline form.

53. THE WILKINSON POWER DIVIDER

54. THE WILKINSON POWER DIVIDER