Design Article
Choosing and using resistive power splitters and dividers
Tony Lymer, Satori Technology
4/7/2008 3:10 PM EDT
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:
where R0 is the system characteristic impedance, and R the resistance seen looking into the network. The transfer characteristics, S21 etc, are equal to the voltage gains measured at the network terminals. So, a 6dB loss translates to an S21 of 0.5. While these numbers are normally complex, the S-parameters of ideal resistive networks all have zero imaginary parts.
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].
ai is the incident wave on port i of the three-port network.
bi is the emerging wave on port i of the three-port network.
From the flow-graph, or from the definition of the S matrix:
Here, |al|2 =|bg|2 is the power arriving at the load, and |bl|2 = |ag|2 is the power leaving or reflected from the load. From Figure 5:
If we compare the terms in equation (13) and equation (15)
Figure 6 shows the overall flow diagram for the three-port network. The important outcome of this analysis is that the apparent or equivalent output impedance of the network is different from the actual output impedance, S22. In the case of port 2, it has the value (S22-S21.S32/S31).
It is of significance only when the ratio of the wave amplitudes, b2 and b3, at ports two and three is concerned. Applying the result to the two networks in question: the three-resistor network yields an equivalent output reflection-coefficient of (0 - 0.5 * 0.5/0.5) = -0.5, and the two-resistor network has an equivalent output reflection coefficient of (0.25 - 0.25 * 0.5/0.5) = 0. The network that gives the least mismatch uncertainty of the output wave ratio is the two-resistor splitter, and not the more common three-resistor divider.
Applications of a two-resistor power splitter
The two-resistor splitter is used in scalar and vector network analyzers where the applied power is split equally between the network under test and the reference channel and the gain of the network being tested requires the calculation of the ratio of the output wave to the reference channel wave. Another common application is in the calibration of power sensors where the calibration factor is calculated from the ratio of the readings of the reference power meter and the instrument under test connected to port 2 of a two-resistor power splitter in turn, while the applied power level is monitored by a third power meter connected to port 3.
If a leveling loop uses a two-resistor power splitter, then the actual output impedance has the same value as the equivalent output impedance, when the loop gain is sufficiently high [3]. This is because the feedback reduces the output impedance of the driver stage almost to zero ohms. High quality, low VSWR power-sensor calibration oscillators like those incorporated into many RF and microwave power meters use this principle.
Other applications for the two-resistor splitter are in calibrating a spectrum or modulation analyzer or signal generator against a power meter with the aim of improving power measurement accuracy. Here again the relative power levels are compared, even though one is measured in a broadband measuring instrument, the power meter, and the other in a narrowband measuring instrument, the spectrum analyzer.
Measuring the effective match of a power splitter
When calculating mismatch uncertainty using splitters and directional couplers, the value of the effective voltage reflection coefficient needs to be known. While the individual S parameters of the network may be measured, and the effective match calculated, this method is effective only at low frequencies. This is partly due to the fact that the formula for the effective match (Γg=S22-S21.S32/S31) (18) incorporates the subtraction of two relatively large numbers that are close in value, in the case of the two-resistor splitter, because the effective voltage reflection coefficient is quite close to zero.
However, for an accurate result, the relative precision of the two constituents (primarily S22 and S32, since S21/ S31 is close to unity if the network is symmetrical) needs to be very high indeed. In practice, the measurement is complicated by the fact that many splitters are non-insertable devices. For example, they may have all female connectors. The fact that the load presented to the third port, for measurement, is assumed to be perfect, when it may be far from ideal, also causes inaccuracies. Measuring such devices with a network analyzer requires considerable experience, skill and care: the conventional use of a network analyzer is not the optimum measurement method.
One efficient method is to incorporate the splitter into a network analyzer and calibrate the analyzer using the open-short-match (OSM) technique. The effective output match can be extracted from some network analyzers as it is calculated as part of the calibration procedure. This method is due to Juroshek [3], and an application note published by Rohde & Shwarz [4] explains the method in more detail. Some of the measurements made of the value of the effective match may be significantly better than the manufacturer's overall specifications, particularly at frequencies much lower than the upper limit for the device. It is reasonable to use these measured values for uncertainty calculations, in place of the manufacturer's specifications, since they give a more realistic estimate of the mismatch uncertainty.
Conclusion
Use a two-resistor power splitter when the output power values are mathematically divided, such as in a measurement of S21 with a network analyzer, a power sensor calibration. In addition, a two-resistor splitter is used in a leveling loop. Use a three-resistor power divider if the mathematical division does not happen, for example, the simultaneous use of a power sensor and a matched frequency counter.
References
[1] Pozar, D.M., Microwave Engineering. (3rd Edition) New York. Wiley 2005.
[2] Powell, R.C.; Banning, H.W.; and Byloff, J.R. "An Automated Power Meter Calibration System". Microwave Symposium Digest, MTT-S International Vol. 82, Issue 1. Jun 1982 pp 357 " 359.
[3] Juroshek, J.R. "A Direct Calibration Method for Measuring Equivalent Source Mismatch" Microwave Journal, October 1997. pp106-118.
[4] Jger, H. "Measurement Method for Determining the Equivalent Reflection Coefficient of Directional Couplers and Power Splitters" Application Note 1EZ51_1E Rohde & Schwarz. 2002.
[5] Engen, G.F. "Amplitude Stabilization of a Microwave Source". IRE Transactions on Microwave Theory and Techniques, Vol. 6, Issue 2, April 1958. pp202-206
About the Author
Anthony Lymer received his B.Sc. (Hons) in Electrical and Electronics Engineering from the University of Wales in 1975 and then joined the Marconi Research Laboratories. From 1977-1981 Tony developed a single-sideband UHF mobile radio system at the University of Bath and joined Hewlett-Packard (now Agilent Technologies) in 1982, working on a wide range of measurement instruments including Carrier-Noise, BER and jitter testers, RF power meters, analogue and digital video testers, signal generators and wireless test-sets. He joined Satori Technology in 2006, and contributed to the design of the ST series power meters. Mr. Lymer is a Member of Institution of Engineering and Technology and a Chartered Engineer and has published more than 30 technical and research articles.
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