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ENE 428 Microwave Engineering Lecture 12 Power Dividers and Directional Couplers 1 Power dividers and directional couplers Passive components that are used for power division or combining. The coupler may be a three-port or a four-port component Three-port networks take the form of T-junctions Four-port networks take the form of directional couplers and hybrids. Hybrid junctions have equal power division and either 90 or a 180 phase shift between the outport ports. 2 Types of power dividers and directional couplers T-junction power divider Resistive divider Wilkinson power divider Bethe Hole Coupler Quadrature (90) hybrid and magic-T (180) hybrid Coupled line directional coupler 3 Basic properties of dividers and couplers P1 Divider or coupler P2aP1 P1=P2+P3 P3(1-a)P1 Divider or coupler P2 P3 The simplest type is a T-junction or a three-port network with two inputs and one output. The scattering matrix of an arbitrary three-port network has nine independent elements S11 S S21 S31 S12 S 22 S32 S13 S 23 S33 4 The scattering parameters’ lossless property The unitary matrix: S t -1 S This can be written in summation form as N S S ki kj ij , k 1 for all i, j where ij = 1 if i = j and ij = 0 if i j thus if i = j, N S ki S ki 1, k 1 while if i j , N S ki S kj 0. k 1 5 It is impossible to construct a three-port lossless reciprocal network. (1) If all ports are matched, then Sii = 0, and if the network is reciprocal the scattering matrix reduces to 0 S S12 S13 S12 0 S 23 S13 S 23 . 0 If the network is lossless, the scattering matrix must be unitary that leads to 6 It is impossible to construct a three-port lossless reciprocal network. (2) S12 S13 1, (1a) S12 S 23 1, (1b) S13 S 23 1, (1c) S13 S23 0, (1d ) S23 S12 0, (1e) S12 S13 0. (1 f ) 2 2 2 2 2 2 Two of the three parameters (S12, S13, S23) must be zeros but this will be inconsistent with one of eq. (1a-c), implying that a three-port network cannot be lossless, reciprocal, and matched at all ports. 7 Any matched lossless three-port network must be nonreciprocal. (1) The [S] matrix of a matched three-port network has the following form: 0 S S21 S31 S12 0 S32 S13 S 23 . 0 If the network is lossless, [S] must be unitary, which implies the following: 8 Any matched lossless three-port network must be nonreciprocal. (2) S12 S13 1, (2a ) S 21 S 23 1, (2b) S31 S32 1, (2c) S31 S32 0, (2d ) S 21 S 23 0, (2e) S12 S13 0. (2 f ) 2 2 2 2 2 2 Either of these followings can satisfy above equations, or S12 S23 S31 0, S21 S32 S13 1, (3a ) S21 S32 S13 0, S12 S23 S31 1. (3b) 9 Any matched lossless three-port network must be nonreciprocal. (3) This results show that Sij Sji for i j, therefore the device must be nonreciprocal. These S matrices represent two possible types of circulators, forward and backward. 10 A lossless and reciprocal three-port network can be physically realized if only two of its ports are matched. (1) If ports 1 and 2 are matched ports, then 0 S S12 S13 S12 0 S 23 S13 S 23 . S33 To be lossless, the following unitary conditions must be satisfied: 11 A lossless and reciprocal three-port network can be physically realized if only two of its ports are matched. (2) S12 S13 1, (3a) S12 S23 1, (3b) 2 2 2 2 S13 S23 S33 1, (3c) S13 S23 0, (3d ) 2 2 2 S12 S13 S23 S33 0, (3e) S23 S12 S33 S13 0. (3 f ) From (3a-b), S13 S23 , so (3d) shows that S13 = S23 = 0. Then |S12|=|S33|=1. 12 A lossless and reciprocal three-port network can be physically realized if only two of its ports are matched. (3) The scattering matrix and signal flow graph are shown below. S21=ejq 1 jq 2 S12=e S33=ejf 3 If a three-port network is lossy, it can be reciprocal and matched at all ports. 13 Four-port networks (Directional Couplers) Input 1 2 Isolated 4 3 Input 1 2 Isolated 4 3 Through Coupled Through Coupled Power supplied to port 1 is coupled to port 3 (the coupled port), while the remainder of the input power is delivered to port 2 (the through port) In an ideal directional coupler, no power is delivered to port 4 (the isolated port). 14 Basic properties of directional couplers are described by four-port networks.(1) 0 S 12 S S 13 S14 S12 S13 0 S 23 S 23 0 S 24 S34 S14 S 24 . S34 0 The [ S ] matrix of a reciprocal four-port network matched at all ports has the above form. If the network is lossless, there will be 10 equations result from the unitary condition. 15 Conditions needed for a lossless reciprocal four-port network (1) The multiplication of row 1 and row 2, and the multiplication of row 4 and row 3 can be arranged so that S14 ( S13 - S 24 ) 0. 2 2 (4) The multiplication of row 1 and row 3, and the multiplication of row 2 and row 4 can be arranged so that S 23 ( S12 - S34 ) 0. 2 2 (5) If S14 = S23 = 0, a directional coupler can be obtained. 16 Conditions needed for a lossless reciprocal four-port network (2) Then the self-products of the rows of the unitary [S] matrix yield the following equations: S12 S13 1, (6a) S12 S24 1, (6b) S13 S34 1, (6c) S 24 S34 1, (6d ) 2 2 2 2 2 2 2 2 which imply that |S13|=|S24|and that |S12|=|S24|. 17 Symmetrical and Antisymmetrical coupler (1) The phase references of three of the four ports are chosen as S12 = S34 = a, S13 = ejq, and S24 = ejf, where a and are real, and q and f are phase constants to be determined. The dot products or rows 2 and 3 gives S12 S13 S24 S34 0 which yields a relation between the remaining phase constant as q + f = 2n. 18 Symmetrical and Antisymmetrical coupler (2) If 2 is ignored, we yield 1. The symmetrical coupler: q = f = /2. 0 a S j 0 a 0 0 j j 0 0 a 0 j . a 0 2. The antisymmetrical coupler: q = 0, f = . 0 a S 0 a 0 0 0 0 - a 0 - . a 0 19 Symmetrical and Antisymmetrical coupler (3) The two couplers differ only in the choice of the reference planes. The amplitudes a and are not independent, eq (6a) requires that a2 + 2 =1. Another way for eq. (4) and (5) to be satisfied is if |S13|=|S24| and |S12|=|S34|. If phase references are chosen such that S13=S24=a and S12=S34=j, two possible solutions are given. First S14=S23=0, same as above. The other solution is for a = =0, which implies S12=S13=S24=S34=0, the case of two decoupled two-port network. 20 Directional coupler’s characterization (1) Power supplied to port 1 is coupled to port 3 (the coupled 2 2 port) with the coupling factor S13 . The remainder of the input power is2 delivered to port 2 (the 2 2 S a 1 . through port) with the coefficient 12 In an ideal coupler, no power is delivered to port 4 (the isolated port). Hybrid couplers have the coupling factor of 3 dB or a = = 1/ 2. The quadrature hybrid coupler has a 90 phase shift between ports 2 and 3 (q = f = /2) when fed at port 1. 21 Directional coupler’s characterization (2) P1 = -20log dB, P3 P dB, Directivity = D = 10 log 3 = 20log P4 S14 P1 Isolation = I = 10 log = -20log|S14| dB. P4 Coupling = C = 10 log The coupling factor indicates the fraction of the input power coupled to the output port. The directivity is a measure of the coupler’s ability to isolate forward and backward waves, as is the isolation. These quantities can be related as I = D + C dB. 22 Ideal coupler The ideal coupler would have infinite directivity and isolation (S14 = 0). 23 The T-junction power divider The T-junction power divider can be implemented in any type of transmission line medium. 24 Lossless divider (1) + V0 - Z0 Z1 jB Z2 Yin A lumped susceptance, B, accounts for the stored energy resulted from fringing fields and higher order modes associated with the discontinuity at the junction. In order for the divider to be matched to the input line impedance Z0, and assume a TL to be lossless, we will have 1 1 1 Yin . Z1 Z 2 Z 0 25 Lossless divider (2) The output line impedances Z1 and Z2 can then be selected to provide various power division ratios. In order for the divider to be matched to the input line impedance Z0, and assume a TL to be lossless, we will have 26 Ex1 A lossless T-junction power divider has a source impedance of 50 . Find the output characteristic impedances so that the input power is divided in a 3:1 ratio. Compute the reflection coefficients seen looking into the output ports. 27 Resistive divider A lossy three-port divider can be made to matched at all ports, although the two output ports may not be isolated. Port 2 P1 Z0/3 Z0/3 Port 1 + V1 - Z0 Zin P2 + VZ - + V2 Z0/3 + V3 - Z0 Z0 Port 3 P3 28 The Wilkinson power divider The lossless T-junction divider cannot be matched at all ports and does not have any isolation between output ports. The resistive divider can be matched at all ports but the isolation is still not achieved. The Wilkinson power divider can be matched at all ports and isolation can be achieved between the output ports. 29