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Transcript

Lecture 6 Chapter 3 Microwave Network Analysis 3.1 Impedance and Equivalent Voltages and Currents 3.2 Impedance and Admittance Matrices 3.3 The Scattering Matrix 3.4 The Transmission (ABCD) Matrix Microwave network analysis Advantages: Extend circuit and network concepts in the lowfrequency circuits to handle microwave analysis and design problems of interest. Basic procedure: (1) Field analysis and Maxwell equations (2) obtain quantities (β, z0, etc.) (3) treat a TL or WG as a distributed component (4) interconnect various components and use network and/or TL theory to analyze the whole system (R, T, Loss, etc.) 3.1 Impedance and Equivalent Voltages and Currents Equivalent voltages and Currents (a) Two-conductor TEM line (b) Non-TEM line Example: TE10 E Quasi-static fields in the transverse surface V= ò + E × dl; I = ò c H × dl; Z 0 = V I Non-uniform V, I along x V= ò b 0 Ey dy = - jwm a p Asin px a e- jb z ò dy y Define equivalent voltages and currents in non-TEM line C1: Voltage and current are defined only for a particular waveguide mode, and are defined so that the voltage is proportional to the transverse electric field, and the current is proportional to the transverse magnetic field. C2: In order to be used in a manner similar to voltages and currents of circuit theory the equivalent voltages and currents should be defined so that their product gives the power flow of the mode. C3: The ratio of the voltage to the current for a single traveling wave should be equal to the characteristic impedance of the line. This impedance may be chosen arbitrarily, but is usually selected as equal to the wave impedance of the line, or else normalized to unity. Transverse fields for an arbitrary WG mode: V(z) = (V +e- jb z +V -e jb z ) + V V e (x, y) + j b z j b z (V e +V e ) (C1 = + = - ) Et (x, y, z) = et (x, y)(A+e- jb z + A-e jb z ) = t A A Apply C1 H t (x, y, z) = ht (x, y)(A+e- jb z - A-e jb z ) = ht (x, y) + - jb z - jb z (I e - I e ) C2 et , ht are the transverse field variations of the mode. C1 I+ I(C2 = + = - ) A A I(z) = (I +e- jb z - I -e jb z ) Apply C2 Complex Power flow of the incident wave which should be equal to ½V+I+*: 1 +2 V + I +* + * * +I+* = ½V P = A òò e ´ h × zds = e ´ h × zds òò 2 2C1C2* s s C1C2* = òò e ´ h s * × zds Integrate over the cross section of the WG Apply C3 Characteristic Impedance Z0 of a WG V + V - C1 Z0 = + = - = I I C2 If choose Z0 = Zw (ZTE or ZTM), C1/C2 = Zw. If choose Z0 = 1 (normalized), C1/C2 =1. Together with C1C2* = òò e ´ h* × zds , C1 and C2 can be solved for a WG mode. Then, s equivalent voltages and currents are defined + + V(z) = (V +e- jb z +V -e jb z ) V = A C1, V = A C1 I(z) = (I +e- jb z - I -e jb z ) I + = A+C2 , I - = A-C2 Propagation constant β and the field intensity A+, A- are calculated from the field analysis. Equivalent voltages and currents for higher WG modes Higher order modes can be treated in the same way, so that a general field in a waveguide can be expressed in the following form: Vn+ - jbn z Vn- jbn z Et (x, y, z) = å( e + e )en (x, y) C1n n=1 C1n N I n+ - jbn z I n- jbn z H t (x, y, z) = å ( e e )hn (x, y) C2n n=1 C2n N where V± and I±, are the equivalent voltages and currents for the n-th mode, and C1n, and C2n arc the proportionality constants for each mode. Example -Equivalent voltage and current for TE10 mode of a rectangular WG Waveguide fields Transmission line model Ey = (A+e- jb z + A-e jb z )sin(p x / a) Hx = -1 + - jb z (A e - A-e jb z )sin(p x / a) ZTE P+ = -1 ab * E H dx dy = A ò y x 2 S 4ZTE + 2 V + = A+C1, V - = A-C1 + P = ab A + 2 4ZTE 1 + +* 1 + 2 = V I = A C1C2* 2 2 V + C1 If choose Z0 = ZTE, then = = ZTE + I C2 Vz = V +e- jb z +V -e jb z I z = I + e- j b z - I - e jb z = 1 + - jb z (V e -V -e jb z ) Z0 1 P = V + I +* 2 I + = A+C2 , I - = A-C2 C1 = C2 = ab 2 1 ZTE ab 2 Impedance (1) Intrinsic impedance of the medium: h = m / e (dependent only on the material parameters of the medium, and is equal to the wave impedance for plane waves.) (2) Wave impedance: Zw = Et / Ht =1/Yw (a characteristic of the particular type of wave. TEM, TM, and TE waves each have different wave impedances, which may depend on the type of line or guide, the material, and the operating frequency.) (3) Characteristic impedance: Z0 =1/ Y0 = V0 / I 0 (The ratio of voltage to current for a traveling wave on a transmission line; uniquely defined for TEM waves; defined in various ways for TE and TM waves. Reflection of a rectangular WG discontinuity for TE10 εr =2.54 a = 2.286cm; b = 1.016 cm Characteristic impedance Reflection coefficient 3.2 Impedance and Admittance Matrices Port: any type of transmission line or transmission line equivalent of a single propagating waveguide mode (one mode, add one electric port). tN: terminal plane providing the phase reference for V, I The total voltage and current on the n-th port: Vn = Vn+ + VnI n = I n+ - I nAn arbitrary N-port microwave network • The impedance (admittance) matrix [Z] ([Y]) of the microwave network relating these voltages and currents é V ê 1 ê V2 ê ê êë VN ù é Z ú ê 11 ú ê Z 21 ú=ê ú ê úû êë Z N1 é I ê 1 ê I2 ê ê êë I N ù é Y ú ê 11 ú ê Y21 ú=ê ú ê úû êë YN1 Z12 Y12 Z1N ùé I1 úê úê I 2 úê úê Z NN úûêë I N ù ú ú ú ú úû [V]=[Z][I] Y1N ùé V1 úê úê V2 úê úê YNN úûêë VN ù ú ú ú ú úû [I]=[Y][V] -1 [Z] =[Y ] [Z] or [Y] is complex number and totally 2N2 elements but many networks are either reciprocal (symmetric matrix) or lossless (purely imaginary matrix), or both and the total number will be substantially reduced. Find the impedance matrix elements Vi Zij = Ij I k =0 for k¹ j Transfer impedance, Zij (between ports i and j): Calculated by driving port j with the current Ij, open-circuiting all other ports (so Ik = 0 for k ≠ j), and measure the open-circuit voltage at port i. Input impedance, Zii Seen looking into port i when all other ports are open-circuited. Find the admittance matrix elements Ii Yij = Vj V =0 k for k¹ j Transfer admittance, Yij (between ports i and j): Calculated by driving port j with the voltage Vj, short-circuiting all other ports (so Vk = 0 for k ≠ j), and measure the short-circuit current at port i. Input admittance, Yii Seen looking into port i when all other ports are short-circuited. Example: Evaluation of impedance parameters Find the Z parameters of the two-port T-network shown below. Solutions: Z11 can be found as the input impedance of port 1 when port 2 is open-circuited: Z11 = V1 = Z A + ZC I1 I2 =0 Similarly, we have Z22 = V2 I2 = Z B + ZC I1=0 The transfer impedance Z12 can be found measuring the open-circuit voltage at port 1 when a current I2 is applied at port 2. By voltage division Z12 = V1 V ZC = 2 = ZC I 2 I1=0 I 2 Z B + ZC It can be verified that Z21 = Z12, indicating the circuit is reciprocal. 3.3 The scattering Matrix The scattering matrix S relates the voltage waves incident on the ports to those reflected from the ports. For some components and circuits, the scattering parameters can be calculated using network analysis techniques. Otherwise, the scattering parameters can be measured directly with a vector network analyzer é ê V1 ê Vê 2 ê ê Vë N ù é ú ê S11 ú ê S21 ú=ê ú ê ú ê SN1 û ë S12 é S1N ùê V1+ ú úê V2+ úê úê SNN úûê VN+ ë ù ú ú ú ú ú û - + [V ]=[S][V ] [S] is symmetric for reciprocal network and unitary for lossless network. Find the admittance matrix elements Vector network analyzer - Vi Sij = + Vj (directly measure S-parameters) Vk+ =0 for k¹ j Sij is found by driving port j with an incident wave of voltage V+ , and measuring the reflected wave amplitude V-, coming out of port i. The incident waves on all ports except the jth port are set to zero, which means that all ports should be terminated in matched loads to avoid reflections. Conversion between Scattering and impedance parameters For a N-ports network, assuming every port has the same impedance and set Z0n = 1, we have Vn = Vn+ + VnI n = I n+ - I n- = Vn+ - Vn- [Z ][I ] = [Z ][V + ] -[Z ][V - ] = [V ] = [V + ]+[V - ] Þ ([Z ] +[U ])[V - ] = ([Z ] -[U])[V + ] Þ [S] = [V - ][V + ]-1 = ([Z ] +[U ])-1 ([Z ] -[U]) 1 0 Identity matrix [U ] = [ ] 0 1 3.4 The transmission (ABCD) matrix Define a 2 x 2 transmission, or ABCD matrix, for each two- port network according to the total voltages and currents The ABCD matrix of the cascade connection of two or more two-port networks can be found by multiplying the ABCD matrices of the individual two-ports. V1 = AV2 + BI 2 I1 = CV2 + DI 2 é V ù é ê 1 ú=ê A êë I1 úû ë C é V ù B ê 2 ú D ûêë I 2 ù ú úû ABCD matrix for a cascaded connection of two-port network é V ù é A ê 1 ú=ê 1 êë I1 úû êë C1 B1 ùé V2 úê D1 úûêë I 2 é V ù é A ê 1 ú=ê 1 êë I1 úû êë C1 ù ú úû é V ê 2 êë I 2 B1 ùé A2 úê D1 úûêë C2 ù é A ú=ê 2 úû êë C2 B2 ùé V3 ù ú úê D2 úûêë I 3 úû B2 ùé V3 úê D2 úûêë I 3 é ù é For a N sections of two- ê V1 ú = ê A1 êë I1 úû êë C1 port networks B1 ù ú D1 úû ù ú úû (The order of ABCD matrix cannot be changed) é A ê N -1 êë CN -1 BN -1 ùé VN úê DN-1 úûêë I N ù ú úû The ABCD parameters of some Useful two-Port circuits Conversion between two-port network parameters