Power dividers and directional couplers

Power dividers and directional couplers are passive devices used mostly in the field of radio technology. They couple a defined amount of the electromagnetic power in a transmission line to a port enabling the signal to be used in another circuit. An essential feature of directional couplers is that they only couple power flowing in one direction. Power entering the output port is coupled to the isolated port but not to the coupled port. A directional coupler designed to split power equally between two ports is called a hybrid coupler.
Directional couplers are most frequently constructed from two coupled transmission lines set close enough together such that energy passing through one is coupled to the other. This technique is favoured at the microwave frequencies where transmission line designs are commonly used to implement many circuit elements. However, lumped component devices are also possible at lower frequencies, such as the audio frequencies encountered in telephony. Also at microwave frequencies, particularly the higher bands, waveguide designs can be used. Many of these waveguide couplers correspond to one of the conducting transmission line designs, but there are also types that are unique to waveguide.
Directional couplers and power dividers have many applications. These include providing a signal sample for measurement or monitoring, feedback, combining feeds to and from antennas, antenna beam forming, providing taps for cable distributed systems such as cable TV, and separating transmitted and received signals on telephone lines.

Notation and symbols

The symbols most often used for directional couplers are shown in figure 1. The symbol may have the [|coupling factor] in dB marked on it. Directional couplers have four ports. Port 1 is the input port where power is applied. Port 3 is the coupled port where a portion of the power applied to port 1 appears. Port 2 is the transmitted port where the power from port 1 is outputted, less the portion that went to port 3. Directional couplers are frequently symmetrical so there also exists port 4, the isolated port. A portion of the power applied to port 2 will be coupled to port 4. However, the device is not normally used in this mode and port 4 is usually terminated with a matched load. This termination can be internal to the device and port 4 is not accessible to the user. Effectively, this results in a 3-port device, hence the utility of the second symbol for directional couplers in figure 1.
Symbols of the form;
in this article have the meaning "parameter P at port a due to an input at port b".
A symbol for power dividers is shown in figure 2. Power dividers and directional couplers are in all essentials the same class of device. Directional coupler tends to be used for 4-port devices that are only loosely coupled – that is, only a small fraction of the input power appears at the coupled port. Power divider is used for devices with tight coupling and is usually considered a 3-port device.

Parameters

Common properties desired for all directional couplers are wide operational bandwidth, high directivity, and a good impedance match at all ports when the other ports are terminated in
matched loads. Some of these, and other, general characteristics are discussed below.

Coupling factor

The coupling factor is defined as:
where P₁ is the input power at port 1 and P₃ is the output power from the coupled port.
The coupling factor represents the primary property of a directional coupler. Coupling factor is a negative quantity, it cannot exceed for a passive device, and in practice does not exceed since more than this would result in more power output from the coupled port than power from the transmitted port – in effect their roles would be reversed. Although a negative quantity, the minus sign is frequently dropped in running text and diagrams and a few authors go so far as to define it as a positive quantity. Coupling is not constant, but varies with frequency. While different designs may reduce the variance, a perfectly flat coupler theoretically cannot be built. Directional couplers are specified in terms of the coupling accuracy at the frequency band center.

Loss

The main line insertion loss from port 1 to port 2 is:
Insertion loss:
Part of this loss is due to some power going to the coupled port and is called coupling loss and is given by:
Coupling loss:
The insertion loss of an ideal directional coupler will consist entirely of the coupling loss. In a real directional coupler, however, the insertion loss consists of a combination of coupling loss, dielectric loss, conductor loss, and VSWR loss. Depending on the frequency range, coupling loss becomes less significant above coupling where the other losses constitute the majority of the total loss. The theoretical insertion loss vs coupling for a dissipationless coupler is shown in the graph of figure 3 and the table below.

Coupling	Insertion loss
dB	dB
3	3.00
6	1.25
10	0.458
20	0.0436
30	0.00435

Isolation

Isolation of a directional coupler can be defined as the difference in signal levels in dB between the input port and the isolated port when the two other ports are terminated by matched loads, or:
Isolation:
Isolation can also be defined between the two output ports. In this case, one of the output ports is used as the input; the other is considered the output port while the other two ports are terminated by matched loads.
Consequently:
The isolation between the input and the isolated ports may be different from the isolation between the two output ports. For example, the isolation between ports 1 and 4 can be while the isolation between ports 2 and 3 can be a different value such as. Isolation can be estimated from the coupling plus return loss. The isolation should be as high as possible. In actual couplers the isolated port is never completely isolated. Some RF power will always be present. Waveguide directional couplers will have the best isolation.

Directivity

Directivity is directly related to isolation. It is defined as:
Directivity:
where: P₃ is the output power from the coupled port and P₄ is the power output from the isolated port.
The directivity should be as high as possible. The directivity is very high at the design frequency and is a more sensitive function of frequency because it depends on the cancellation of two wave components. Waveguide directional couplers will have the best directivity. Directivity is not directly measurable, and is calculated from the addition of the isolation and coupling measurements as:
Note that if the positive definition of coupling is used, the formula results in:

S-parameters

The S-matrix for an ideal symmetrical directional coupler is given by,
In general, and are complex, frequency dependent, numbers. The zeroes on the matrix main diagonal are a consequence of perfect matching – power input to any port is not reflected back to that same port. The zeroes on the matrix antidiagonal are a consequence of perfect isolation between the input and isolated port.
For a passive lossless directional coupler, we must in addition have,
since the power entering the input port must all leave by one of the other two ports.
Insertion loss is related to by;
Coupling factor is related to by;
Non-zero main diagonal entries are related to return loss, and non-zero antidiagonal entries are related to isolation by similar expressions.
Some authors define the port numbers with ports 3 and 4 interchanged. This results in a scattering matrix that is no longer all-zeroes on the antidiagonal.

Amplitude balance

This terminology defines the power difference in dB between the two output ports of a hybrid. In an ideal hybrid circuit, the difference should be. However, in a practical device the amplitude balance is frequency dependent and departs from the ideal difference.

Phase balance

The phase difference between the two output ports of a hybrid coupler should be 0°, 90°, or 180° depending on the type used. However, like amplitude balance, the phase difference is sensitive to the input frequency and typically will vary a few degrees.

Transmission line types

Directional couplers

Coupled transmission lines

The most common form of directional coupler is a pair of coupled transmission lines. They can be realised in a number of technologies including coaxial and the planar technologies. An implementation in stripline is shown in figure 4 of a quarter-wavelength directional coupler. The power on the coupled line flows in the opposite direction to the power on the main line, hence the port arrangement is not the same as shown in figure 1, but the numbering remains the same. For this reason it is sometimes called a backward coupler.
The main line is the section between ports 1 and 2 and the coupled line is the section between ports 3 and 4. Since the directional coupler is a linear device, the notations on figure 1 are arbitrary. Any port can be the input, which will result in the directly connected port being the transmitted port, the adjacent port being the coupled port, and the diagonal port being the isolated port. On some directional couplers, the main line is designed for high power operation, while the coupled port may use a small connector, such as an SMA connector. The internal load power rating may also limit operation on the coupled line.
Accuracy of coupling factor depends on the dimensional tolerances for the spacing of the two coupled lines. For planar printed technologies this comes down to the resolution of the printing process which determines the minimum track width that can be produced and also puts a limit on how close the lines can be placed to each other. This becomes a problem when very tight coupling is required and couplers often use a different design. However, tightly coupled lines can be produced in air stripline which also permits manufacture by printed planar technology. In this design the two lines are printed on opposite sides of the dielectric rather than side by side. The coupling of the two lines across their width is much greater than the coupling when they are edge-on to each other.
The λ/4 coupled-line design is good for coaxial and stripline implementations but does not work so well in the now popular microstrip format, although designs do exist. The reason for this is that microstrip is not a homogeneous medium – there are two different mediums above and below the transmission strip. This leads to transmission modes other than the usual TEM mode found in conductive circuits. The propagation velocities of even and odd modes are different leading to signal dispersion. A better solution for microstrip is a coupled line much shorter than λ/4, shown in figure 5, but this has the disadvantage of a coupling factor which rises noticeably with frequency. A variation of this design sometimes encountered has the coupled line a higher impedance than the main line such as shown in figure 6. This design is advantageous where the coupler is being fed to a detector for power monitoring. The higher impedance line results in a higher RF voltage for a given main line power making the work of the detector diode easier.
The frequency range specified by manufacturers is that of the coupled line. The main line response is much wider: for instance a coupler specified as might have a main line which could operate at. The coupled response is periodic with frequency. For example, a λ/4 coupled-line coupler will have responses at nλ/4 where n is an odd integer. This preferred response gets obvious when a short impulse on the main line is followed through the coupler. When the impulse on the main line reaches the coupled line a signal of the same polarity is induced on the coupled line similar to the response of an RC-high-pass. This leads to two non-inverted pulses on the coupled line that travel in opposite direction to each other. When the pulse on the main line leaves the coupled line an inverted signal is induced on the coupled line, triggering two inverted impulses that travel in opposite direction to each other. Both impulses on the coupled line that go in the same direction as the pulse on the main line are of opposite polarity. They cancel each other so there is no response on the exit of the coupled line in forward direction. This is the decoupled port. The pulses on the coupled line that travel in the opposite direction to the pulse on the main line are also of opposite polarity to each other but the second impulse is delayed by twice the delay of the parallel line. For a λ/4 coupled-line the total delay length is λ/2 so the second signal is inverted and this gives a maximum response on the coupled port.
A single λ/4 coupled section is good for bandwidths of less than an octave. To achieve greater bandwidths multiple λ/4 coupling sections are used. The design of such couplers proceeds in much the same way as the design of distributed-element filters. The sections of the coupler are treated as being sections of a filter, and by adjusting the coupling factor of each section the coupled port can be made to have any of the classic filter responses such as maximally flat, equal-ripple, or a specified-ripple response. Ripple is the maximum variation in output of the coupled port in its passband, usually quoted as plus or minus a value in dB from the nominal coupling factor.
It can be shown that coupled-line directional couplers have purely real and purely imaginary at all frequencies. This leads to a simplification of the S-matrix and the result that the coupled port is always in quadrature phase with the output port. Some applications make use of this phase difference. Letting, the ideal case of lossless operation simplifies to,