Chapter 7Power Dividers and Couplers 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 DividerThree Port Network S ParameterThe scattering matrix of an arbitrary three port network has nine independent variables:[ ]111]1

33 32 3123 22 2113 12 11S S SS S SS S SSIf 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.ji ijS S Three Port Network S ParameterIf all the three ports are matched the [S] matrix can be written as:[ ]111]1

00023 1323 1213 12S SS SS SSIf the three port network is not reciprocal then and its [S] matrix will not be symmetric. ji ijS S Power Dividers and CouplersIn 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 SplitterPower CombinerPower Dividers and CouplersTwo basic types of these circuits are: Power Divider and Power Combiner.1 < 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.Directional CouplerOne 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 CouplerInputIsolatedThroughCoupledUsually 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).Directional CouplerDirectional CouplerInputIsolatedThroughCoupledDirectional 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.

,_

31log 10PPC

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43log 10PPD

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41log 10PPIdB C D I + Hybrid CouplersHybrid 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:[ ]1111]1

0 1 01 0 00 0 10 1 021jjjjS[ ]1111]1

0 1 1 01 0 0 11 0 0 10 1 1 021SThe Lossless T JunctionThe 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:1 3 21 1 1Z Z Z +The Lossless T JunctionIn 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 3.1 1 3 2 1P P P P P + + 1 + 211212222121ZZZVPZVPo o 311213232121ZZZVPZVPo o The Lossless T JunctionInput matching requirement:( )( )0||||3 1 2 1 3 23 1 2 1 3 21 3 21 3 2111+ + ++ Z Z Z Z Z ZZ Z Z Z Z ZZ Z ZZ Z ZZ ZZ ZLL3 23 21 3 2 1 3 2) (Z ZZ ZZ Z Z Z Z Z+ + 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?The Lossless T JunctionNow lets consider matching of port 2.( )( )3 2 2 1 3 23 2 2 1 3 22 3 12 3 1222||||Z Z Z Z Z ZZ Z Z Z Z ZZ Z ZZ Z ZZ ZZ ZLL+ + ++ If port 2 is matched, then 2 = 0 and we will have:3 13 12 3 1 2 3 10 ) (Z ZZ ZZ Z Z Z Z Z+ + Substitution ofyields Z2 = 0.Which shows that the lossless T junction cannot be matched at all three ports simultaneously. (Problem 7.1 Pozar) 3 23 21Z ZZ ZZ+Junction T ExampleIf 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. The resistive power divider for an equal power split.The Resistive DividerAssuming that all the lumped-element resistors are terminated in the characteristic impedance Zo, the input impedance looking into any port is:oo oin oooo oinZZ ZZ ZZZZ ZZ +

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+

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+ + 323 3 3 3Since the network is symmetric from all three ports, the output ports are also matched. Therefore, S11=S22=S33=0The Resistive DividerThe voltage at the center of the junction is:1 13232332VZ ZZV Vo oo+The output voltages V2 and V3 are equal to:1003 221433V VZZZV V Vo + Scattering Matrix of the Resistive DividerSince 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:[ ]111]1

0 1 11 0 11 1 021SoinZVP2121( )ino oPZVZVP P4181212121213 2 Half of the supplied power is dissipated in the resistors.