Lecture 3

1. Lecture 3 • Impedance and Equivalent Voltages and Currents for Non-TEM Lines • Impedance Properties of One-Port Networks • Impedance, Admittance and Scattering Matrices • Signal Flow Graphs Microwave Techniques

2. Impedance and Equivalent Voltages and Currents for Non-TEM Lines • ·show how circuit and network concepts can be extended to handle many microwave analysis and design problems of practical interest Microwave Techniques

3. Impedance and Equivalent Voltages and Currents for Non-TEM Lines • equivalent voltage and current can be defined uniquely for TEM-type lines (require two conductors) but not so for non-TEM lines Microwave Techniques

4. Impedance and Equivalent Voltages and Currents for Non-TEM Lines • for non-TEM lines, voltage and current are only defined for a particular waveguide mode, V is related to Et and I to Ht where t denotes the transverse component Microwave Techniques

5. Impedance and Equivalent Voltages and Currents for Non-TEM Lines • the product of the equivalent V and I should yield the power flow of the mode • V/I for a single traveling wave should be equal to the characteristic impedance of the line or can be normalized to 1 Microwave Techniques

6. Impedance and Equivalent Voltages and Currents for Non-TEM Lines • for an arbitrary waveguide mode with a +ve and -ve (in z) traveling waves Microwave Techniques

7. Impedance and Equivalent Voltages and Currents for Non-TEM Lines • we can write the voltage and current of an equivalent transmission line as Microwave Techniques

8. Impedance and Equivalent Voltages and Currents for Non-TEM Lines • i.e., we are only interested in certain quantities and these quantities can be derived using circuit and network theory Microwave Techniques

9. Impedance and Equivalent Voltages and Currents for Non-TEM Lines • the incident power is given by • Where Microwave Techniques

10. Impedance and Equivalent Voltages and Currents for Non-TEM Lines • the characteristic impedance of the equivalent transmission line is Microwave Techniques

11. Impedance and Equivalent Voltages and Currents for Non-TEM Lines • the wave impedance is given by Microwave Techniques

12. Impedance and Equivalent Voltages and Currents for Non-TEM Lines • if we choose the characteristic impedance of the line equal to that of the wave impedance, i.e., • which can be either TE or TM modes Microwave Techniques

13. Impedance and Equivalent Voltages and Currents for Non-TEM Lines • therefore, if one can measure the voltage and current for each mode, the field in the waveguide can be determined as sum of the field for each mode Microwave Techniques

14. Impedance and Equivalent Voltages and Currents for Non-TEM Lines Microwave Techniques

15. Impedance and Equivalent Voltages and Currents for Non-TEM Lines • we reiterate that there are various types of impedance • intrinsic impedance • depends on material parameters but is equal to the wave impedance of a plane wave in a homogeneous medium Microwave Techniques

16. Impedance and Equivalent Voltages and Currents for Non-TEM Lines • wave impedance of a particular type of wave, namely, TEM, TE and TM • depends on frequency, materials and boundary conditions Microwave Techniques

17. Impedance and Equivalent Voltages and Currents for Non-TEM Lines • characteristic impedance • unique for TEM waves, non-unique for TE and TM Microwave Techniques

18. Impedance and Equivalent Voltages and Currents for Non-TEM Lines • we will calculate the wave impedance of the TE10 mode waveguide • now consider the field equations for the TE10 rectangular waveguide mode, the field components are Microwave Techniques

19. Impedance and Equivalent Voltages and Currents for Non-TEM Lines Microwave Techniques

20. Impedance and Equivalent Voltages and Currents for Non-TEM Lines Microwave Techniques

21. Impedance and Equivalent Voltages and Currents for Non-TEM Lines • compare with the transmission line equations and the incident power Microwave Techniques

22. Impedance and Equivalent Voltages and Currents for Non-TEM Lines Microwave Techniques

23. Impedance and Equivalent Voltages and Currents for Non-TEM Lines • therefore, we can relate the field and circuit parameters for the TE10 waveguide mode • we can also the transverse resonance technique to look at the wavenumber of the TE10 mode in the y direction (height of the waveguide) Microwave Techniques

24. Impedance and Equivalent Voltages and Currents for Non-TEM Lines • the waveguide can be regarded as a transmission line with certain characteristic impedance and is shorted at both ends Microwave Techniques

25. Impedance and Equivalent Voltages and Currents for Non-TEM Lines • using the transmission line equation to transfer the short circuit to y =y' • according to the transverse resonance technique, we have • impedance looking downward + impedance looking upward = 0 Microwave Techniques

26. Impedance and Equivalent Voltages and Currents for Non-TEM Lines • for the above equation to be true for any value of y', ky must be zero • this is correct as ky = np/b where n = 0 for TE10 mode Microwave Techniques

27. Impedance Properties of One-Port Networks • consider the arbitrary one-port network shown here Microwave Techniques

28. Impedance Properties of One-Port Networks • assume that • if then Microwave Techniques

29. Impedance Properties of One-Port Networks • the input impedance • for a lossless network, R = 0, therefore Microwave Techniques

30. Impedance Properties of One-Port Networks • the input impedance is purely imaginary • the reactance is positive for an inductive load (Wm > We) and is negative for a capacitive load Microwave Techniques

31. Impedance Properties of One-Port Networks • Foster s reactance theorem: the rate of change of the reactance for the lossless one-port network with frequency is Microwave Techniques

32. Impedance Properties of One-Port Networks • from Maxwells equations • and the vector identity Microwave Techniques

33. Impedance Properties of One-Port Networks Microwave Techniques

34. Impedance Properties of One-Port Networks Microwave Techniques

35. Impedance Properties of One-Port Networks • Note that • and therefore, • =0 Microwave Techniques

36. Impedance Properties of One-Port Networks • V= j XI for lossless line Microwave Techniques

37. Impedance Properties of One-Port Networks • poles and zeros must alternate in position as the slope is always positive Microwave Techniques

38. Even and Odd Properties of Z(w) and G(w) • define the Fourier transform as • note that v(t) must be a real quantity, i.e., v(t) = v*(t), therefore, • or V(-w) = V*(w) Microwave Techniques

39. Even and Odd Properties of Z(w) and G(w) • note that we can only measure V(w), we need its complex conjugate to obtain v(t), similar arguments hold for I(w) • V*(-w) = V(w)=Z(w)I(w)=Z*(- w)I*(- w)=Z*(-w)I(w) • Z(w) =Z*(- w) Microwave Techniques

40. Even and Odd Properties of Z(w) and G(w) • therefore, the real part of Z, i.e, R is an even function of w while the imaginary part X is an odd function of w • the reflection coefficient G also has an even real part and an odd imaginary part Microwave Techniques

41. Even and Odd Properties of Z(w) and G(w) Microwave Techniques

42. Impedance, Admittance and Scattering Matrices Microwave Techniques

43. Impedance, Admittance and Scattering Matrices • N-port microwave network, each port has a reference plane tn • at the reference plane of port N, we have Microwave Techniques

44. Impedance, Admittance and Scattering Matrices • if we are only interested in knowing the relationship among the voltages and currents at the ports, we can define a impedance matrix Z so that Microwave Techniques

45. Impedance, Admittance and Scattering Matrices • the element Zij of the impedance matrix is given by • similar equations can be written for the admittance matrix  Microwave Techniques

46. Impedance, Admittance and Scattering Matrices • for reciprocal network, the impedance (admittance) matrix is symmetric • for lossless network, all matrix elements are purely imaginary Microwave Techniques

47. Impedance, Admittance and Scattering Matrices • the scattering matrix relate the voltage waves incident on the ports to those reflected from the ports • the scattering parameter is written as Microwave Techniques

48. Impedance, Admittance and Scattering Matrices • each element is given by • Sij is found by driving port j with an incident wave of voltage Vj+, and measuring the reflected amplitude coming Vi-, out of port i Microwave Techniques

49. Impedance, Admittance and Scattering Matrices • the incident waves on all ports except the jth port are set to zero which implies that all these ports are terminated with match loads • for a reciprocal network • [S] = [S]t, i.e., the matrix is symmetric Microwave Techniques

50. Impedance, Admittance and Scattering Matrices • for a lossless network • for all I,j • the scattering parameters can be readily measured by a Network Analyzer Microwave Techniques