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Rat-Race Couplers and Their S-Parameter Matrices

Key Takeaways

  • A directional coupler is a reciprocal multi-port network that gives two amplitude outputs when an input signal is given.  

  • Rat-race couplers are hybrid, ring-directional couplers. 

  • From a rat-race coupler’s S-parameter matrix, the antisymmetric property can be identified.

Directional coupler

A directional coupler is a reciprocal multi-port network used in RF circuits

Hybrids and couplers are indispensable components in RF and microwave applications. These components are used in balanced amplifiers, frequency discriminators, balanced mixers, and various other RF systems.

Hybrids are formed by connecting circuit elements directly. The sections of transmission lines placed in proximity form couplers that have four ports with matching characteristics. The low value of the reflection coefficient over the application frequency range makes a coupler suitable for RF systems.

Rat-race couplers are commonly-used directional couplers for RF and microwave systems. In rat-race couplers, S-parameters are used for finding the dependency of the characteristic impedance on the insertion loss, coupling, power splits, etc. In this article, we will discuss rat-race couplers and their S-parameters.  

Directional Couplers and Their Characteristics

A directional coupler is a reciprocal multi-port network that gives two amplitude outputs when an input signal is given. In directional couplers, there are four ports: input, direct, coupled, and isolated. 

Let’s imagine a directional coupler with the input port fed with power P1 and powers available at the direct, coupled, and isolated ports (P2, P3, and P4, respectively). This directional coupler exhibits important characteristics such as:

  1. Coupling - The coupling factor (C) is given by the ratio of input port power (P1) to the coupled port power (P3) in decibels, provided reflectionless terminations terminate all the ports.

Expression for coupling factor

  1. Directivity - Directivity (D) is calculated as the ratio of the coupled port power (P3) to the isolated port (P4) in decibels, provided reflectionless terminations terminate all the ports. 

          Expression for directivity

  1. Isolation - Isolation (I) is calculated as the ratio in decibels of power at an isolated port power (P4) to the input port power (P1) in decibels.

Expression for isolation

  1. Matching - The voltage standing-wave ratio (VSWR) or the input or output port reflection coefficient provides the matching in the directional coupler, provided reflectionless terminations terminate all the ports. 

  2. Insertion loss - The ratio of input port power (P1) and output port power (P4) in decibels is called insertion loss, provided reflectionless terminations terminate all the ports. 

  3. Power split - The difference between the coupling and insertion loss in decibels gives the power split characteristics of the directional coupler. 

  4. Bandwidth - The range of frequencies for which a parameter falls within a specified limit with respect to certain characteristics. 

  5. Phase balance - The relative phase difference of output waves is given by phase balance. The most popular directional couplers offer phase balances of 0° (in-phase directional couplers), 90° (quadrature directional couplers), and 180° (out-of-phase directional couplers).

Let’s examine the most popular directional coupler: the rat-race coupler. 

Rat-Race Couplers

Rat-race coupler configuration

Rat-race couplers are hybrid couplers with a coupling factor of 3 dB. They are manufactured using microstrips in the shape of a ring or circle. The three branches of a rat-race coupler are 90° phase shifted from each other, and the other branch is 270° phase-shifted. In equally split rat-race couplers, the input power is split equally between port 2 and port 3; port 4 is kept isolated. In branch-line rat-race couplers, the output signals have a phase difference of 180° between each other.

Considerations When Using Rat-Race Couplers

  • Two in-phase combined input signals can be added using rat-race couplers with minimal to no loss.

  • Rat-race couplers can split the input signal equally and deliver it through two ports with zero phase difference.

  • These couplers can be used as power dividers with a 180° phase shift.

  • Rat-race couplers can be used as power combiners that add two signals with 180° phase shifts and minimal loss.

Advantages of Rat-Race Couplers

  • Wide bandwidth

  • Acceptable return loss 

  • Good isolation 

  • Amplitude or phase imbalance is offered in a frequency range from 2.20 GHz to 3.30 GHz. 

The S-Parameter Matrix of a Rat-Race Coupler

A scattering parameter matrix gives a clear description of the type of coupler. For a reciprocal coupler matched at all ports, S-parameters are given by the matrix below.

Coupler S-parameter matrix

When the phase constants θ and ɸ are 0 and 180°, respectively, the coupler is said to be an antisymmetric coupler with an S-parameters matrix of:

Antisymmetric coupler S-parameter matrix

Magic Tee Coupler

The magic tee coupler is another significant hybrid coupler that offers a 180° phase difference between ports 2 and 3 when input is fed at port 4. The scattering matrix of such a rat-race coupler can be given by:

Magic Tee matrix

In summary, the rat-race coupler is an antisymmetric coupler. From a rat-race coupler’s S-parameter matrix, antisymmetric properties can be identified. Cadence’s suite of PCB design and analysis tools can assist you with the design of different directional couplers, including rat-race and magic tee couplers. Cadence offers design and analysis tools that authenticate the design of the rat-race couplers in your RF systems.

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