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Microwave Network Analysis We have shown in our study of transmission lines that circuit analysis techn iques are appli cable to transmi ssion lines car rying TEM waves. These circuit anal ysis techn iques may be app lied to lines car rying TEM waves, like the coaxial line below, since a unique cu rrent and v oltage can be defined at any point along the transmission line. The capability to define a unique voltage (as a line integral of the el ectric field) and current (as a line integral of the magnetic field) for the TEM wave is directly related to the fact that the transverse fields of the TEM wave are equivalent to the electrostatic and magnetostatic fields for the same c onductor geometry. These field characteristics are also true for the general two-condu ctor transmission line carrying a TEM wave. The definitions of the current and vol tage on the TEM line are independent of the integral paths chosen (the voltage integral may originate at any point on the outer condu ctor and terminate at any point on t he inner condu ctor while the magnetic field integral path may take on any shape as long as the closed p ath encloses the inner con ductor while lying within the o uter condu ctor). The se circuit analysis techniques are also applicable to wave guiding structures that employ qua si-TEM waves (microstrip) since the transver se fields ar e essentially the same a s pure TE M waves. For wave guiding structures that cannot suppo rt a TEM wave (non- TEM lines) like a r ectangular waveg uide, we ca nnot define a unique voltage and current at a given point along the st ructure. It works ou t that the value of the defined voltage and current will depend on the integral path chosen (there are an infinite number of possible currents and voltages). If the current and v oltage are not unique, then there is also no unique impedance in the circuit analysis sense (a ratio of voltage to current). For these r easons, we choo se to define equivalent voltages, currents and impedances for non-TE M lines which, even t hough they ar e not unique, yield the proper physical behavior of the guided wave (power flow, attenuation, etc.). The following are the rules that we use in the definition of these equ ivalent paramet ers for non-TE M lines. (1) Equi valent voltages, currents and impedances are defined for each non-TEM mod e. (2) The equivalent voltage is defined to be propor tional to the transverse electric field. (3) The equivalent current is defined to be propor tional to the transverse magnetic field. (4) The produc t of the equivalent voltage and c urrent yields the power flow of the mo de at that point on the non-TEM line. (5) The ratio of the equivalent voltage to the equivalent current defines an equivalent cha racteristic impedance for the non - TEM line. T he choice of the equivalent characteristic impedance is arbitrary, but is normally chosen as either the wave i mpedan ce of t he given mod e, or normalized t o unity. Using these guidelines, we may define the transverse field s of an arbitrary non-TE M mod e on a gener al wave gu iding structure as where e(x,y) and h(x,y) are vectors defining the transverse variation of the transver se fields, and the con stants A and A are the field amplitudes of the+ ! forward and reverse traveling wav es, respectively. Note that the wav e coefficients and the current and v oltage constants are related by The transver se vectors e(x,y) and h(x,y) are related by the particular wave impedance of the given mode. We have shown for general TE and TM modes on a wave gu iding structure that so that the transver se vectors e(x,y) and h(x,y) are related by w TM TE where Z is the wave im pedance (either Z or Z). If we define the equivalent characteristic impedance for the given mode of the waveguide as the wave impedance of the mode, then Alternatively, we may normalize the equ ations and choose 12 A second eq uation for the u nknown co nstants C and C may be found by enforcing the power flow condition. The complex power flow in the + z direction a long the waveguide may be defined us ing t he corresponding Poynting vector. The power flow in the + z direction is found by integrating the Poy nting vector over the cr oss-section of the w aveg uide (S). According to circuit theory, the power flow in the equivalent circuit should be(1a) (1b) so that We may solve equations (1) and ( 2) simultaneously to de termine the 12 coeffi cients C and C. Th is process i s repeat ed for each of the mo des within the w aveg uide to yield the gener al expression for the total transver se fields in terms of t he equ ivalent voltages and currents.(2) Example (Waveguide equivalent voltage and current) Consider the fields and wav e impedance associated with the dom inant 10TE mod e in a r ectangular waveg uide. Note th at th ese solutions contain only a f orward traveling wave. If we include waves t raveling in both directions, we may write the transver se fields w ithin the w aveg uide as 10 where the tran sverse variati on of the TE mode transverse field s are defined by t he fun ctions: There are currents and vol tages associated with the rectangul ar waveguide 10TE mode which are defined by The voltage coeffi cients (V and V) and current coeff icients (I and I) are+ ! + ! related to the fi eld coefficients A and A by+ ! 12 where the con stants C and C are given by where the surf ace S is the cross-sectional surface of the waveguide. If we choos e the equivalent characteristic impedance of the waveguide to the be 10 the w aveg uide T E wave i mpedan ce, then Evaluation of the integral in the second equation for the unknown constants yields (2) 12 Solving equations (1) and (2) for C and C yields(1) 10 The e quivalent waveguide current and vo ltage for the TE mode become Note that the equations above represent an equivalent transmission line model for the w aveg uide. Example (Waveguide discontinuity, equivalent transmission line model) Consider a rectangul ar waveguide which is air-filled ov er a portion of the w aveg uide (z < 0) and di electric-filled ove r the remaining portion of 10 the w aveg uide (z > 0). Assume the the dom inant TE mode is propagating in the air-filled portion of the waveguide. Determine the fields in both portions of the waveguide using the transmission line equivalent model. We may employ the equivalent voltage and c urrent equations for the 10 rectangular waveguide TE mode and model the waveguide discontinuity as a connection of two transmissi on lines w ith different ch aract eristic impedances. 10 Given the incident TE mod e in the ai r-filled portion of the w aveg uide, part of the wave is transmi tted into the dielectric-filled po rtion o f the waveguide while the remainder of the wave is ref lecte d back into the air- filled region. The equivalent voltage in the two regions of the waveguide may be w ritten as According to the equivalent transmi ssion line mod el, the ratio of the reverse traveling wave vo ltage coeffi cient to that the of the forward wave i s equal to the reflection coefficient of the transmission line connection. where There is no r everse traveling wav e in the dielectric-filled region a nd the ratio of the forward wave voltage coefficient in the dielectric region to the forward wave vol tage coefficient in the air region i s equal to the transmission coefficient for the transmission line conne ction. The correspond ing t ransverse fields within the two regions of the waveguide are determined according to Microw ave One-Port Network A general microwave one -port network is defined by a device for which po wer can enter or leave through onl y one transmission line or waveg uide. We assume t hat the one-port network i s defi ned by a sur face S which is per fectly con ducting except for an opening at the terminal conne ction. We may apply the general form of Poy nting’s theorem t o de scribe the power flow for the o ne-port network. Th e gener al form of P oynting’s theorem for the closed surface S shown below is sP -power delivered by the sources within S. oP -power passing o utward through S. lP -power dissipated within S. mW -magnetic energy stored within S. eW -electric energy stored within S. For the one-port network, we assume that no s ources are located within the s surface S (P = 0). The unit normal n is an inward pointing normal for the o one- port network suc h that the term P is negative and represents the power flow into the one-port network. Poynting’s th eorem for the one-port network beco mes We may define the transver se fields over the w ave gu iding structure as Note that the terminal voltage V and current I at th e input to the one-port network is The p ower flow into the one-port network i s then given by t he surf ace integral o f the tran sverse field s in the term inal plane (opening). Note that the equ ation above is a restatement of the previously ob tained power flow relationship for the general wave gu iding structure: The power flo w can be related to the input impedance of the one-port network which according to circuit theory is Since t he v oltage and cur rent that we ar e usi ng in this impedan ce relationship may be the equivalent voltage and current of a waveguide, the resulting input impedance would be an equivalent impedance. The power flow equation can be r ewritten as so that the input impedance of the one-port network is These equations show t hat the resistance o f the one-port is related to the power dissipated within S while the reactance of the one-port is related to the net reactive energy stored within in S. If the one-port is characterized l by lossless materials, then P = 0 and R = 0. The reactance of the one-port me em is inductive (positive) if W > W or capaci tive (negative) if W > W. Symmetry of the M icrow ave Network Inp ut Impedance and Re flection Coe fficient We kno w from circuit theory that the resistive and r eactive components of impedance have certain symmetry characteristics with respect to frequency. Since ci rcuit theory is simply a l ow-frequency approximation of field theory, we find that th ese im pedance symmetry relationships hold true for microwave network input impedances. If we define the standard Fo urier transfor m pair for the time-domain voltage v(t) and the corr espon ding frequency- domain voltage V(ù), we have The time-domain voltage must be real such that v(t) = v(t) which gives* If we make the change of variable from ù to !ù in the integral fo r v(t), we* find so that the fr equency- domain voltage mu st satisfy which means t hat Re{V(ù)} must be even wi th respect to ù while Im{V(ù)} must be odd with respect to ù. The impedan ce can be written as The real term abo ve [R(ù)] must be even since it is defined in terms of products o f even functions and products o f odd functions. The imaginary term [X(ù)] must be o dd since i t is defi ned in term s of products of even functions and odd functions. Thus, The real and imaginary portions of the reflection c oefficient at the input of the one-port network also exhibit symmetry with respect to ù. The reflection coefficient as a function o f frequency is defined by Evaluating this expression at !ù yields Z(-@)-Z, R(@)-Z,-jX(o which shows that Re{I'(@)} =even with respect to Im{T()} =oddwith respect to N-Port Microw ave Network The general N-port microwave net work i s shown below where N is the total num ber of ports. The ports may be fed by any combination o f transmission lines or waveguides. We assume that each wave guiding structure carries only the single dominant mode. A terminal plane (transverse plane) is defined for each port where the equivalent voltage and n current will be d efined. The terminal plane i s desi gnated as t for the nth port. Discontinuities in the guiding structure will g enerally generate evanescent modes. If we choos e the terminal planes far enoug h away from these discontinuities, then the evanescent modes decay sufficiently to be neglected. If the coordinates of the wave guiding structures are chosen such that the terminal planes are each located at z = 0, then the voltage and current at the n terminal plane may be w ritten asth Note that the reverse wave t raveling out of the n port is dependent on theth the reflection from the n port and waves that are coupled into the n portth th from the other ports. T hus, the impedance of the overall N-port network must be defined by an impedance matrix [Z] such t hat or The individual element s of t he impedance mat rix may b e det ermined according to j In other words, we may drive port j with a current I while open-circuiting all other ports except j and meas ure the resulting open-circuit respon se at port i. The ratio of the open-circuit voltage at port i to the current at p ort ij j gives us the impedance matrix element Z. We may also define an admittance matrix according to or The individual element s of the admittance mat rix may b e det ermined according to j Thus, we may drive port j with a vol tage V while short-circuiting all other ports except j and meas ure the resulting short-circuit current respon se at port i. The ratio of the short-circuit current at port i to the voltage at port ij j gives us the admittance matrix element Y. According to the d efinition of t he i mpedan ce a nd admi ttance matrices, these mat rices are inverses so that If the N-port microwave network is passive (no sources) and contains only isotropic media, the network is a reciprocal net work and b oth the impedance and admittance matrices are symmetric. Examples of anisotropic m aterials (parameters are functions of direction - tensor ì and/or å) are ferrites and plasmas. If the network is lossless, then the impedan ce and admi ttance mat rices are purely imagi nary. Scattering Matrix The equ ivalent currents and voltages used t o define the impedan ce and admittance matrices for the general N-port network are somew hat abstr act in that they cann ot be easi ly meas ured for a gi ven network at microwave frequencies. However, we may easily measure the amplitude and phase angle of the wave reflected (or scattered) from a port relative to the amplitude and phase angle of the wave incident on that port. Thus, we define a scattering matrix which relates the scattered voltage coefficients (V) to the incident wave vo ltage coeffi cients (V) according to! + or The individual elements of the scattering m atrix may be determined according to Thus, we may launch an incident wave toward port j while all other ports have no incident waves ( the transmi ssion lines or waveg uides on these ports should be terminated by a ma tched load) and meas ure the scat tered wave at port i. The ratio of the scattered wave at port i to the incident wave ij at port j gives us the scattering matrix element S. The elements of the scattering matrix are referred to as t he s-parameters of the network. Examp le (Determination of s- paramet ers) Determine the s-parameters for the 2-port network characterized by the series connection of transmission lines and a lumped reactance shown below. 11 21 Determination of S, S (excite port #1, matched termination on port #2) Note that the matched termin ation elimin ates any “incident” wave on port 2 #2 (V = 0).+ For the series reactance conne ction, we may relate the two port currents by If we assume t hat both ports are located at a coordi nate reference of z = 0 for each transmi ssion line, then the curr ent relation can be w ritten as 22 12 Determination of S, S (excite port #2, matched termination on port #1) The matched termination eliminates any “incident” wave on p ort #1 1(V = 0).+ Again, we may relate the two po rt currents and find The overall scattering matrix for the two-port network is Properties of the Scattering Matrix If we norma lize all ports of the N-port microwave network to the same characteristic impedance, then For a rec iprocal networkY [S] is symmet ric For a lo ssless networkY [S] is unitary The mat rix [S] is unitary if it satisfies [S] [S] = [U]t* where [S] = the transpose of [S]t [S] = the conjugate of [S]* [U] = identity matrix For a unitary matrix [S], the produc t of any column of [S] with the conjugate of that column gives unity. The produc t of any column of [S] with the conjugate of any other column gives zero. Scattering Matrix in Terms of the Impedance Matrix If we assume that th e characteristi c im pedances of all N-ports are on identical and choose this characteristic impedan ce to be unity (Z= 1), then When these equ ations for the voltage and current vectors are incorporated into the impedance matrix definition, we find Grouping the terms involving the forward voltage coeffi cients and reverse voltage coeffi cients yields, According to the definition o f the scattering m atrix, so that the scattering matrix in terms of the impedance matrix is We can also solve this equation for the impedance matrix in terms of the scattering m atrix whi ch yields S-Pa ram eters at Arbitrary Termi nal Planes We have assumed that all terminal planes for the wave guiding structures connected to the N-port microwave network are located at z = 0. If we wish to sh ift the terminal planes t o so me ar bitrary locations at n distances l away from the z = 0 reference, a new scattering matrix must be determined. Note that the terminal planes have been moved away from the N-port network. Shifting the planes closer to the N-port would require the opposite sign on the z-coordinates. The incident and scattered v oltage w aves at the o riginal and shifted terminal planes for the N ports are related by different scattering matrices. If we deno te all quantities at the shi fted terminal planes w ith a pr ime, then we may write According to the general equations for the equivalent voltage as a function of position on a wave gu iding structure, the voltage on the n port is giventh by Thus , the coefficients of the incident and scattered vo ltage waves at the original terminal planes ( unprimed t erms) and the shi fted terminal planes (primed terms) are related by In mat rix form, the coeffi cients of the incident and scattered voltage w aves are related by Inserting these incident and scattered wave vectors into the definition o f the scat tering matrix gives Solving the equ ation above for the scat tered w ave coeffi cients gives where The equation for the scattered wave coefficients defines the scattering matrix at the shi fted terminal planes [SN] in terms of the scattering matrix at the original terminal planes [ S]. This equation shows that there is a characteristic phase shift [defined by the electrical length associated wi th the phy sical length of the terminal plane nn n shift, (è = âl)] for the incident and scattered waves. That is, the incident waves reach the shifted terminal plane before the original plane, and the scattered w aves r each the shi fted terminal plane aft er the original plane. The Tran smission M atrix (ABCD Matrix) Many microwave ne twork pr oblems invol ve series conne ctions (cascading) of several two-port networks. For this type of network, it is conveni ent to de fine a special set of two-port parameters kno wn as the transmission parameters. The transmission parameters (defined by A,B,C and D) make up the transmission matrix which is also cal led the ABCD matrix. The transmi ssion matrix of a network formed by sever al cascaded two-ports is simply the produc t of the transmi ssion matrices of the individual two-port s. The transmission m atrix of a given two-port network relates the input voltage and current to the output voltage and current according to 2 Note that the convention for the direction of the output current (I) has been changed from our previous conve ntion. Th is change in the current direction allows one to equate the output current of one stage to the input current of the following stage. In matrix form, the transmission equations are where Given a pair of cas cade d two-port networks as shown be low, the transmi ssion matrices for the network #1 and network #2 can be d efined as Simple substitution yields which de fines the input qua ntities of the ove rall network to the output quantities. This techn ique is easily extended to the ser ies connection of an arbitrary number of two-port networks. General ized S cattering Param eters The incident and scattered waves in the definition of the N-port network sca ttering par amet ers may be n orma lized so that the p ower delivered to each p ort is indepen dent of t he cha racteristic impedan ces. Using the scattering mat rix de finition, the voltage and c urrent at the terminal plane of t he n port (z = 0) isth Assuming the char acteristic impedan ces of the N ports are real, the power delivered to the n port isth If we define a new set of incident and scattered wave coefficien ts for port nn N (a and b) according to then the voltage and current at the terminal plane of t he n port areth The power delivered to the n port in terms of the new wave coefficientsth is The generalized scattering matrix [S] relates the normalized incident and nn scattered w ave coeffi cients a and b. or ij where the element S is defi ned as The elements of the generalized scattering ma trix are related to scattering matrix by where defines the correspond ing sc attering mat rix term. Examp le (previous s-paramet er examp le) The scattering m atrix for this example was found to be Note that the scattering matrix for this reciprocal network i s not symmet ric. The scattering m atrix for this configuration woul d be symmetric if the characteristic impedan ces of the two transmi ssion lines were equal. We can show that th e generalized scattering matrix is sy mmetric for N-port networks with ports of different characteristic impedance. According to the transformation of the scattering m atrix to the generalized scattering m atrix: The g eneralized sca ttering mat rix is sym metric (given the r eciprocal network) even though the characteristic imp edances of the two ports are unequal. Equivalent Circuits for T wo-Po rt Networks Once the two-port paramet ers (Z, Y, S, T) for a given microwave device have been determined, we need a circuit configuration which is equivalent to the defining equations of the respect ive two-port paramet ers. If the mi crowave dev ice is reciprocal , the equ ivalent circuit can be d efined in terms o f six independent parameters (the real and imaginary parts of three unique matrix elements). There are an unlimited number of equivalent circuit configurations that are possible. Tw o commo nly used configurations are the T-net work in terms of impedance parameters and the ð-network in terms o f admittance parameter s. These n etworks are eas ily implemented with any type of two-port parameters given the basic transformations among the different sets of parameters (see Table 4.2, p. 211) T-Network m-Network q L, a<_ Fi5 : —+ Port#1 Port#2 TV *¥id+ VN) =MN +YM 1,=VV +Yin) #VM,~VaM-M2)=YoY+Yoa Signal Flow Graphs Signal flow graphs are a graphical technique of analyzing m icrowave networks in terms of the incident and scattered (transmitted and reflected) waves at the network po rts. Signal flow graphs consist of nodes and branches which may be related directly to the generalized scattering paramet ers. Nodes -Each node represents either an incident wave ( a wave entering the port) or a scattered wave (a wave leaving the port). Thus, th e signal flo w graph for a an N-port network contains 2N node s. Following the conve ntion of the g eneralized sca ttering par amet ers, the i port isth i defined by two nodes: a represents the wave entering the i i port while b represents the w ave l eaving the i port.th th Branches -Each branch is a path from an a-node to a b-node which represents the signal flow within the N-port network. Thus , each br anch is associated wi th a particular scattering parameter or a reflection coefficient. Decompos ition Rul es for Signal Flow Graphs Series Rule Parallel Rule Self-loop Rule Splittin g Rule Exampl e (Signal flow graph) D etermine the input reflection in coefficien t (Ã) for the terminated two-port network shown be low using a signal flow graph. The input reflection c oefficient may be determined by manipulating the signal flow graph for the terminated two-port network. We need si gnal graph m odels for the conne ction o f the source to the two-port and the connection of the load to the two-port. The sig nal flow graph for the source in the terminated two-port network must include reflections due to mismatch. The signal flow graph m odel for the input source (shown below) accounts for scattered waves from the two port network that may be reflected ba ck to the network through the s reflection coefficient for the source, Ã. Note that the input voltage has been no rmalized according to the scattering p arameter definition. The signal flow graph for the termination accounts for reflections due to l mismatch through the load reflection coefficient, Ã. Combining the signal flo w graphs of the source, the load and the two-port network yields the signal flow graph for the complete circuit. 22 2 We may use the splitting rule on node a. The pa th directly from a to b can be t ransfor med i nto a sel f-loop by noting that 2 Node b can t hen be transfor med u sing the sel f-loop rule. 2 12 1 The path from node a to b to a to b can the be combined into a single path using the ser ies rule. 11 The two pat hs between no des a and b can be comb ined using the parallel rule. 11 The si ngle pat h from node a to b allows us to wr ite the input reflection coefficient directly from the reduced signal flow graph.