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Lecture 7

Lecture 7. Power Divider Quadrature (90 o ) Hybrid Coupled Line Directional Couplers The 180 o Hybrid. Resistive Divider. a three-port power divider can be matched at all ports using lumped resistors consider the circuit diagram below:. Resistive Divider.

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Lecture 7

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  1. Lecture 7 • Power Divider • Quadrature (90o) Hybrid • Coupled Line Directional Couplers • The 180o Hybrid Microwave Technique

  2. Resistive Divider • a three-port power divider can be matched at all ports using lumped resistors • consider the circuit diagram below: Microwave Technique

  3. Resistive Divider • the input impedance ZI at V is equal to where Z is the impedance looking into a Zo/3 resistor followed by a 50 W transmission line the factor of 2 is due to two parallel lines of equal impedance Microwave Technique

  4. Resistive Divider • the input impedance at V1 is therefore given by • Which is matched to the transmission line Microwave Technique

  5. Resistive Divider • due to symmetry, all 3 ports are matched, i.e., • input power at Port 1 will be equally divided between Port 2 and Port 3 Microwave Technique

  6. Resistive Divider • if the voltage at Port 1 is equal to V1, the voltage V at the junction is equal to Microwave Technique

  7. Resistive Divider • from voltage division again, the voltages at Port 2 and Port 3 are • the transmission from Port 1 to 2 is therefore given by Microwave Technique

  8. Resistive Divider • Due to symmetry, • the scattering matrix is given by Microwave Technique

  9. Resistive Divider • note that the matrix is reciprocal due to symmetry, it is not a unitary matrix due to the resistive loss • the input power at Port 1 is given by • while the output power at Port 2 and Port 3 are both • , half of the input power is dissipated by the three resistors Microwave Technique

  10. The Wilkinson Power Divider • note that the S23 and S32 in the resistive divider are nonzero, i.e., input power from Port 2 can be coupled to Port 3 and vice versa • It can be shown that the Wilkinson power divider can be matched at all ports with port isolation, i.e., S23 and S32 are both zero Microwave Technique

  11. The Wilkinson Power Divider • the Wilkinson power divider can be made to give arbitrary power division, however, we will concentrate on the equal power division Microwave Technique

  12. The Wilkinson Power Divider • it is convenient to normalize the characteristic impedance to 1 so that the Wilkinson power divider circuit is given by Microwave Technique

  13. The Wilkinson Power Divider • note that the transmission line at Port 1 is replaced by two parallel resistors with a normalized value of 2 each Microwave Technique

  14. The Wilkinson Power Divider • it will be shown that Z is equal to and r=2 • to analyze this circuit, it is convenient to employ the even and odd symmetry • the final answer is obtained by combining the results from even- and odd-mode analysis Microwave Technique

  15. Even Mode Analysis • when Vg2=Vg3, there is no current going through the resistor r/2 as V2 and V3 have the same potential; therefore, these resistors can be removed Microwave Technique

  16. Even Mode Analysis • we can simplify the circuit by only consider half of the circiut Microwave Technique

  17. Even Mode Analysis • looking into Port 2, we have • Patch 2 is matched, when and therefore Z = ; here the transmission line acts as a quarter-wave transformer Microwave Technique

  18. Even Mode Analysis • all the input power at Port 2 will be delivered to Port 1, i.e., S22 = 0 • to find S12, let us consider the transmission line Microwave Technique

  19. Even Mode Analysis • The voltage alone the line is given by • at x = 0, V(x) =V2 and at x=l/4, V=V1 • the reflection coefficient G is given by • and Microwave Technique

  20. Even Mode Analysis • substituting Z = , we have Microwave Technique

  21. Even Mode Analysis • due to symmetry, we also have • and • From voltage division, Microwave Technique

  22. Odd Mode Analysis • when Vg2=-Vg3, the voltage will change from Vg2 at Port 2 to -Vg2 at Port 3 • the voltage must be zero at the point on the plane of symmetry Microwave Technique

  23. Odd Mode Analysis • we can simplify the circuit by grounding the circuit at the plane of symmetry Microwave Technique

  24. Odd Mode Analysis • looking into Port 2, we have a short-circuited l/4 line in parallel with a r/2 resistor, the input impedance reads • Port 2 is matched when and therefore, r=2; here the transmission line converts a short circuit to an open circuit Microwave Technique

  25. Odd Mode Analysis • all the input power at Port 2 will be delivered to the r/2 resistor, and none to Port 1, i.e., = 0, due to symmetry, we also have • From voltage division, • the scattering matrix can be obtained from the even- and odd-mode results Microwave Technique

  26. Odd Mode Analysis • since Ports 2 and 3 are matched, they are also zero for both even and odd mode Microwave Technique

  27. The Quadrature (90o) Hybrid • quadrature hybrids are 3 dB directional couplers with a 90o phase difference in the outputs Microwave Technique

  28. The Quadrature (90o) Hybrid • with all the ports matched, power entering Port 1 will be equally divided between Port 2 and Port 3 with a 90o phase difference between the two • no power is coupled to Port 4 Microwave Technique

  29. The Quadrature (90o) Hybrid • the scattering matrix is given by • the scattering matrix can be obtained easily by using even-odd mode analysis Microwave Technique

  30. The Quadrature (90o) Hybrid • the circuit of the 90o hybrid is given below • the actual response can be obtained by the sum of the even and odd excitations Microwave Technique

  31. The Quadrature (90o) Hybrid • At the plan of symmetry, • for even symmetry, the stubs terminate at A and B with an open circuit • for odd symmetry, the stubs terminate at A and B with a short circuit • the length of the stubs are l/8 Microwave Technique

  32. The Quadrature (90o) Hybrid • define the even and odd reflection and transmission coefficients for a two-port network as Ge,o and Te,o respectively • the scattering parameters are given by Microwave Technique

  33. The Quadrature (90o) Hybrid • the analysis is conveniently presented by cascading ABCD matrices Microwave Technique

  34. The Quadrature (90o) Hybrid • the shunt stubs are l/8, the admittance at A is • , tan  = 1 • for even symmetry, YL = 0, YA = j (normalized) • for odd symmetry, YA = -j (normalized) Microwave Technique

  35. The Quadrature (90o) Hybrid • for even mode, the ABCD matrix of the open circuit shunt stub is Microwave Technique

  36. The Quadrature (90o) Hybrid • the ABCD matrix of the series stub is Microwave Technique

  37. The Quadrature (90o) Hybrid • the ABCD matrix from A to B is given by Microwave Technique

  38. The Quadrature (90o) Hybrid • ABCD matrix can be converted into scattering parameters Microwave Technique

  39. The Quadrature (90o) Hybrid • for the odd mode, the ABCD matrix from A to B is given by Microwave Technique

  40. The Quadrature (90o) Hybrid • the scattering parameters are given by • Port 1 is matched, half power transmitted to Port 2 with -90o phase shift • Port 4 isolated, half power transmitted to Port 3 with -180o phase shift Microwave Technique

  41. The Quadrature (90o) Hybrid • due to the quarter-wave transformer, the bandwidth of the 90o hybrid is limited to 10-20% • this design can be modified for unequal power division Microwave Technique

  42. Coupled Line Directional Couplers • when two unshielded transmission lines are close together, power can be coupled between the lines Microwave Technique

  43. Coupled Line Directional Couplers • C11 and C22 are the self capacitance in the absence of the other line • C12 is the mutual capacitance between the two lines in the absence of the ground plane Microwave Technique

  44. Coupled Line Directional Couplers • for the even mode, the electric field has even symmetry and the field lines of one transmission line repel those of the other line, therefore, C12 is effectively open-circuited Microwave Technique

  45. Coupled Line Directional Couplers • the characteristic impedance for the even mode is • for the odd mode, the electric field have an odd symmetry about the symmetry plane and a voltage null exists between the two strip conductors • this is effectively putting a ground plane between the conductors Microwave Technique

  46. Coupled Line Directional Couplers • the effective capacitance between either strip conductor and ground is • the characteristic impedance for the odd mode is • the transmission lines are assumed TEM lines, this is true for stripline but only approximately true for microstrip line Microwave Technique

  47. Coupled Line Directional Couplers • a single-section coupled line coupler is shown below Microwave Technique

  48. Coupled Line Directional Couplers • the input impedance at Port 1 of the coupler is given by Microwave Technique

  49. Coupled Line Directional Couplers • the input impedance for the even and odd modes are given by Microwave Technique

  50. Coupled Line Directional Couplers • by voltage division, we have Microwave Technique

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