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Hanyang University Microwave Engineering by David M. Pozar Ch. 4.1 ~ 4 / 4.6 Woobin Kim Antennas & RF Devices Lab. Hanyang University Contents 4.1 Impedance And Equivalent Voltages and Currents - 4.1.1 Equivalent Voltages and Currents - 4.1.2 The Concept of Impedance - 4.1.3 Even and Odd properties of π π ππ§π πͺ(π) 4.2 Impedance And Admittance Matrices - 4.2.1 Reciprocal Networks - 4.2.2 Lossless Networks 2/41 Hanyang University Contents 4.3 The Scattering Matrix - 4.3.1 Reciprocal Networks and Lossless Networks - 4.3.2 A Shift in Reference Planes - 4.3.3 Power Waves and Generalized Scattering Parameters 4.4 The Transmission (ABCD) Matrix - 4.4.1 Relation to Impedance Matrix - 4.4.2 Equivalent Circuits for Two-Port Networks 3/41 Hanyang University Contents 4.6 Discontinuities and Modal Analysis - 4.6.1 Modal Analysis of an H-plane Step in Rectangular Waveguide 4/41 Hanyang University 4.1 Impedance and Equivalent Voltages and Currents (In waveguide Field) 4.1.1 Equivalent Voltages and Currents β’ At microwave frequencies the measurement of voltage or current is difficult, (or impossible) unless a clearly defined terminal pair is available. The voltage for an arbitrary two-conductor TEM transmission line. The total current flowing on the + conductor (Integration contour is any close path enclosing the + conductor) 5/41 Hanyang University β’ A characteristic impedance π0 can then be defined for traveling waves as π0 = π πΌ . β’ We can proceed to apply the circuit theory for transmission lines as a circuit element. β’ The situation is more difficult for waveguides. For the dominant ππΈ10 modes, the transverse fields can be written, (from Table 3.2) 6/41 Hanyang University β’ Applying to the electric field equation, β’ Thus it is seen that this voltage depends on the position, x and the length of the integration contour along the y direction. No βcorrectβ voltage β’ A similar problem arises with currents, and impedances that can be useful for non-TEM lines. β’ For an arbitrary waveguide mode with both positively and negatively traveling waves. The fields can be written as 7/41 Hanyang University π , β : The transverse field variations π΄+ , π΄β : The field amplitude of the traveling waves β’ Because πΈπ‘ and π»π‘ are related by wave impedance, ππ β’ Defines equivalent voltage and current waves as ( 1 π+ = π + πΌ +β 2 β’ The complex power flow for the incident wave is given by 8/41 ) Hanyang University β’ To make this power to be equal to (1/2)π + πΌ +β , β΅ π + = πΆ1 π΄ , πΌ + = πΆ2 π΄ Characteristic impedance If it is desired to have π0 = ππ , πΆ1 π0 = =1 Desirable to normalize the characteristic impedance, πΆ2 β’ For given waveguide mode, equations can be solved for the πΆ1 , πΆ2 . Higher order modes can be treated in the same way. 9/41 Hanyang University 4.1.2 The Concept of Impedance β’ We used the idea of impedance in several different ways. We summarize the various types of impedance we have used so far, and their notation: β’ π = π/π : Intrinsic impedance of the medium. (Equal to the wave impedance for plane waves) β’ ππ = πΈπ‘ /π»π‘ = 1 /ππ : Wave impedance. (This impedance may depend on the type of line or guide, the material, and the operating frequency) β’ π0 = 1/π0 = π + / πΌ + : Characteristic impedance. (The ratio of voltage to current. This impedance for such waves may be defined in different ways.) 10/41 Hanyang University β’ Consider the arbitrary one-port network shown in above Figure ππ : Average power dissipated by the network ππ , ππ : Stored magnetic and electric energy β’ Define real transverse modal fields π and β , (with normalization) 11/41 Hanyang University β’ Thus we see R, of the input impedance is related to the dissipated power, while X is related to the net energy stored in the network. β’ If the network is lossless, then ππ = π = 0. Then πππ is purely imaginary, with a reactance β’ Which is positive for an inductive load (ππ > ππ ) , and negative for a capacitive load (ππ < ππ ). 12/41 Hanyang University 4.1.3 Even and Odd properties of π π and πͺ(π) β’ At the input port of an electrical network, The voltage and current at this port are related as π π = π π πΌ(π). β’ For an arbitrary frequency dependence, we can fine the time-domain voltage by taking the inverse Fourier transform of π π . β’ Because π£ π‘ = π£ β π‘ , (π£ π‘ must be real) β’ Following above equations, β’ Which means that π π π π is even and πΌπ{π π } is odd. 13/41 Hanyang University β’ Similarly, current must be real too, so πΌ βπ = πΌ β (π). β’ Thus, if π π = π π + ππ π , then π π is even and π π is odd. β’ Now consider the reflection coefficient at the input port: β’ Same result in Ξ π like π(π). Which shows that Ξ(π) 2 , Ξ(π) are even function. Ξ(π) 2 (or Ξ(π) ) = 14/41 Hanyang University 4.2 Impedance and Admittance Matrices β’ This type of representation lends itself to the development of equivalent circuits of arbitrary networks, which will be quite useful when we discuss the design of passive components such as couplers and filters. β’ We begin by considering an arbitrary N-port microwave network 15/41 Hanyang University β’ At a specific point on the n th port, a terminal plane, π‘π , is defined along with equivalent voltages and currents for the incident (ππ + , πΌπ + ) and reflected (ππ β , πΌπ β ) waves. (When z = 0) 16/41 Hanyang University (πππ : Input impedance looking into port i when all other ports are open) (πππ : Transfer impedance between ports i and j when all other ports are open) (πππ : Input admittance looking into port i when all other ports are short) (πππ : Transfer admittance looking between port i and j when all other ports are short) β’ If the network is reciprocal, we will show that the impedance / admittance matrices are symmetric β’ The network is lossless, we can show that all the Z and Y elements are purely imaginary. 17/41 Hanyang University 4.2.1 Reciprocal Networks β’ Let πΈπ , π»π and πΈπ , π»π be the fields anywhere in the network due to two independent sources, a and b located somewhere in the network. β’ The fields due to sources a and b can be evaluated at the terminal planes π‘1 and π‘2 . 18/41 Hanyang University ππ ,βπ : Transverse modal field ππ ,πΌπ : Equivalent total voltages, currents β’ Let πΈπ , π»π and πΈπ , π»π be the fields anywhere in the network due to two independent sources, a and b located somewhere in the network. ( πΈ1π is the transverse electric field at terminal plane π‘1 of port 1 due to source a. ) β’ Substituting the all fields ( E, H ) into above equation, 19/41 Hanyang University β’ As in section 4.1, the power through a given port can be expressed as ππΌ β /2; then, πΆ1 = πΆ2 = 1 for each port, so that β’ Use the 2 X 2 admittance matrix of the two-port network to eliminate the πΌ, β’ Z zzzz can take on arbitrary values. So we must have . We have the general result that β’ Then if [Y] is a symmetric matrix, its inverse, [Z] is also symmetric. 20/41 Hanyang University 4.2.2 Lossless Networks β’ Now consider a reciprocal lossless N-port junction. (Impedance and admittance matrices must be pure imaginary.) β’ The net real power delivered to the network must be zero. β’ We could set all port currents equal to zero except for the nth current, So, 21/41 Hanyang University β’ Let all port currents be zero except for πΌπ , πΌπ . β΄ Purely real quantity (none zero) 4.3 The Scattering Matrix β’ A specific element of the scattering matrix can be determined as 22/41 Hanyang University β’ Then, for convenience, we can set π0π = 1. The total voltage and current at the nth port can be written as π βΆ Unit identity matrix β’ From the above matrix equation, Equal to Reflection coefficient at port 1 23/41 Hanyang University 4.3.1 Reciprocal Networks and Lossless Networks β’ The scattering matrices for these particular types of networks have special properties. For a reciprocal network, the scattering matrix is symmetric. For a lossless network, matrix is unitary. β΅ Comparing with result for 23 page, 24/41 for the reciprocal network Hanyang University β’ If the network is lossless, no real power can be delivered to the network. If the characteristic impedance of all the ports are identical, unity, Total incident power Total reflected power ( 25/41 ) Hanyang University 4.3.2 A Shift in Reference Planes β’ This chapter appear scattering parameters are transformed when the reference planes are moved from their original locations. 26/41 Hanyang University β’ Consider a new set of reference planes defined at ππ = ππ for the nth port, and let the new scattering matrix be denoted as [πβ²]. β’ From the theory of traveling waves on lossless transmission lines we can relate the new wave amplitudes ( : The electrical length of the outward shift of the reference plane of port n) 27/41 Hanyang University β’ πβ²ππ meaning that the phase of πππ is shifted by twice the electrical length of the shift in terminal plane n. β’ This result gives the change in the reflection coefficient on a transmission line due to a shift in the reference plane. 4.3.3 Power Waves and Generalized Scattering Parameters β’ Inverting the total voltage and current on a transmission line in terms of the incident and reflected voltage wave amplitudes, β’ The average power delivered to a load can be expressed as 28/41 Hanyang University β’ But this result is only valid when the characteristic impedance is real. β’ In addition, these result are not useful when no transmission line is present between the generator and load, as in the circuit shown in above figure. β’ It is possible to define a new set of waves, called power waves. (Generally useful concept) The incident and reflected power wave amplitudes To make b = 0, 29/41 Hanyang University β’ Then the power delivered to the load can be expressed as β’ This result that the load power is the difference between the powers of the incident and reflected power waves. β’ This result is valid any reference impedance ππ . (β΅ β’ From basic circuit theory, 30/41 ) Hanyang University β’ With β’ The power delivered to the load is β’ When the load is conjugately matched to the generator, β’ Maximum only when β’ We define the power wave amplitude vectors, ( ) 31/41 Hanyang University β’ Can be written, 4.4 The Transmission (ABCD) Matrix β’ We will see that the ABCD matrix of the cascade connection of two or more two-port networks can be easily found by multiplying the ABCD matrices of the individual two-ports. 32/41 Hanyang University 33/41 Hanyang University 4.4.1 Relation to Impedance Matrix β’ The impedance parameters of a network can be easily converted to ABCD parameters. 34/41 Hanyang University 4.4.2 Equivalent Circuits for Two-Port Networks β’ Because of the physical discontinuity in the transition from a coaxial line to a microstrip line, E,M energy can be stored in the vicinity of the junction, leading to reactive effects. Reference table 4.1, 4.2 in page 190, 192 Purely imaginary (lossless) 35/41 Hanyang University 4.6 Discontinuities and Modal Analysis β’ By either necessity or design, microwave circuits and networks often consist of transmission lines with various types of discontinuities. β’ Depending on the type of discontinuity, the equivalent circuit may be a simple shunt or series element across the line or a T- / pi- equivalent circuit may be required. β’ The component values of an equivalent circuit depend on the parameters of the line and the discontinuity. 4.6.1 Modal Analysis of an H-plane Step in Rectangular Waveguide β’ The technique of waveguide modal analysis is relatively straightforward and similar in principle to the reflection/transmission problems that were discussed in Chapter 1 and 2. 36/41 Hanyang University β’ By either necessity or design, microwave circuits and networks often consist of transmission lines with various types of discontinuities. 37/41 Hanyang University β’ The geometry of the H-plane waveguide step in Figure, It is assumed that only the dominant ππΈ10 mode is propagating in guide 1 (z < 0) and is on the junction from z < 0. The propagation constant of the ππΈπ0 mode in guide 1 The wave impedance of the ππΈπ0 mode in guide 1 β’ Reflected / Transmitted waves in both guides β’ Higher order modes are also important in this problem. Because they account for stored energy 38/41 Hanyang University β’ Not existed ππΈππ modes for π β 0, any TM modes. β’ At z = 0, the transverse fields must be continuous for 0 < x < c; in addition, πΈπ¦ must be zero for c < x < a because of the step. 39/41 Hanyang University β’ Solving for π΄1 ( The reflection coefficient of the incident ππΈ10 mode) β’ After the π΄π are found, the π΅π can be calculated from β’ The equivalent reactance can be found from the reflection coefficient π΄1 from 40/41 Hanyang University β’ The Figure shows the normalized equivalent inductance versus the ratio of the guide widths c/a for a free-space wavelength π = 1.4π and for N = 1,2. 41/41 Hanyang University Thank you for your attention Antennas & RF Devices Lab.