Choosing and using resistive power splitters and dividers

The simple task of splitting an RF signal equally into two paths has several solutions. Choosing the correct method of doing this and the appropriate system component requires some thought : starting with the nomenclature.

Power dividers and splitters are often confused, and they are sometimes referred to as power combiners or couplers, since they are reciprocal networks, and can be applied in reverse. Some vendors refer to the two-resistor networks as splitters and the three-resistor networks as dividers, and this convention is used here.

Pozar [1] lists several other networks that can divide or combine signals, such as Wilkinson power dividers, quadrature hybrid dividers and magic-T hybrid dividers to name but three. Lumped element versions of these networks are used at lower RF, and even audio frequencies. However, to simplify the discussion, this article only deals with the two resistive networks.

The major difference between the resistive networks and those that use reactive components or depend on the size of a resonant transmission-line is that the resistive networks are truly broadband, while the other networks tend to have more limited bandwidths. For example, a resistive power splitter may cover 0 to 40 GHz, while some directional couplers have a 5:1 frequency range. However, the resistive dividers and splitters are lossy, and this is not acceptable in some circumstances, such as high-power applications where the other networks mentioned are used.

The most common form of resistive power divider is the three-resistor form. It also has a "dual," which is a network with a different schematic that is externally indistinguishable from the original. One would have to look inside the package to differentiate the two networks. One version has resistors in the form of a star, and the other in the form of a delta.

A less common power splitter has only two resistors and has an output-impedance that is not equal to the characteristic impedance of the system. This can be confusing as it is used in the most demanding of situations: network analyzers and power sensor calibration where the correct circuit loading is vital. The reason this is acceptable is explained in the following analysis.

Derivation of the S-parameters for the ideal networks is straightforward. Assuming that all ports are loaded in resistors equal to the characteristic impedance, the input or output reflection coefficients, S₁₁, S₂₂ and S₃₃ are given by:

S-Parameters of three-resistor divider

The above network has a dual, which is a network with the same S-parameters, but a different circuit diagram, this is shown below.

Pozar [1] gives the theoretical S-parameter matrix for a perfect example of this network as:

The diagonal of the matrix is made up of zeros thus S₁₁ = S₂₂ = S₃₃ = 0. This implies that the impedance seen looking into the network at any port is Z₀ ohms, provided the other ports are terminated in Z₀ ohms, so, all ports are matched. The gain of the network is | S₂₁|² = 0.25 or 6 dB loss.

S-Parameters of two-resistor splitter

Here, S₂₂ = S₃₃ = 0.25, so the output ports are not matched. Again, the loss of the network in a matched system is 6 dB.

General three-port analysis
The three port network may be inserted between a signal source with output wave bg, reflection coefficient Γ₁ and two load resistors with reflection coefficients Γ₂ and Γ₃. For most of the applications, a power sensor replaces at least one of the load resistors. Engen presented the result of this analysis in [5], and it was later derived by Powell et al [2].

a_i is the incident wave on port i of the three-port network.
b_i is the emerging wave on port i of the three-port network.

From the flow-graph, or from the definition of the S matrix:

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