2. MICROWAVE CIRCUITS CHARACTERIZATION . Introduction . Z-Parameters [POZAR 5.3] [COLLIN 4.5] . Y-Parameters [POZAR 5.4] [COLLIN 4.5] . Transmission (ABCD-) Parameters [POZAR 5.5] [COLLIN 4.9] . Interconnection of two-port networks [POZAR 5.6] . S-Parameters [POZAR 5.4] [COLLIN 4.7] • Cons of [Z] and [Y] parameters • Definition • Properties of S-parameters • Pros of [Z] and [Y] parameters • Reference planes shift • Measurement of S-parameters • Conversions between two-port network parameters [POZAR 5.6] . Two-port networks [POZAR 5.6] [COLLIN 4.11] • Reflection coefficients • Power transfer Radiofrequency Engineering C. Collado, J.M. González-Arbesú 1 EETAC-UPC 2. MICROWAVE CIRCUITS CHARACTERIZATION . S-parameters of some two-port networks • Attenuators and amplifiers •Isolators[POZAR 10.4] • Filters . S-matrix of networks with more than two-ports [POZAR 8.1] • Power dividers [COLLIN 6.6] [POZAR 8.2, 8.3] • Circulators [COLLIN 6.10] [POZAR 10.6] • Directional Couplers [COLLIN 6.4, 6.5] [POZAR 8.5, 8.8] [COLLIN] R.E. Collin, Foundations for Microwave Engineering, Wiley-Interscience, 2nd Edition, 2001 (New York) [POZAR] D.M. Pozar, Microwave Engineering, Addison-Wesley Publishing Company, 2nd Edition, 1993 (Reading, Massachusetts) Radiofrequency Engineering C. Collado, J.M. González-Arbesú 2 EETAC-UPC GLOSSARY -1/2 • an : normalized voltage input wave at port n [V ] •[a] : row vector of normalized voltage input waves at network ports [V-1/2] -1/2 • bn : normalized voltage output wave at port n [V ] •[b] : row vector of normalized voltage output waves at network ports [V-1/2] • : phase constant [rad·m-1] • C : coupling [dB] • D : directivity [dB] • G : power gain [adim] • GA : available gain [adim] • GT : transducer power gain [adim] • I : isolation [dB] • IL : insertion loss [dB] + • In : incident current phasor at port n [A] - • In : reflected current phasor at port n [A] • In : total current phasor at port n [A] •[I] : row vector of current at ports [A] • l : transmission line length [m] • L : attenuation [dB] Radiofrequency Engineering C. Collado, J.M. González-Arbesú 3 EETAC-UPC GLOSSARY • Pavn : power available from a network (assuming conjugate matching) [W] • Pavs : power available from the source (assuming conjugate matching) [W] • Pi : incoming power at work [W] • PIN : power delivered to the input of a network [W] • PL : power delivered to the load [W] • Pincident : incoming power to the port [W] • Preflected : reflected power [W] • Ptransmitted : transmitted power [W] • R : resistance [] • RS : serial resistance [] • RP : shunt resistance [] • RL : return loss [dB] • G : (voltage) source reflection coefficient [adim] • IN : (voltage) reflection coefficient of network input port [adim] • L : (voltage) load reflection coefficient [adim] • OUT : (voltage) reflection coefficient of network output port [adim] • Sij : scattering parameter between nodes i and j [adim] •[S] : scattering matrix [adim] • tn : reference plane at port n Radiofrequency Engineering C. Collado, J.M. González-Arbesú 4 EETAC-UPC GLOSSARY • VCO : voltage-controlled oscillator • VG : voltage at generator [V] + • Vn : incident voltage phasor at port n [V] - • Vn : reflected voltage phasor at port n [V] • Vn : total voltage phasor at port n [V] •[V] : row vector of voltages at ports [V] -1 • Yij : transfer admitance between nodes i and j [S= ] •[Y] : admitance matrix [S] • Zij : transfer impedance between nodes i and j [] •[Z] : impedance matrix [] • ZG : generator impedance [] • ZIN : impedance at the input port of the network [] • ZL : load impedance [] • Z0 : characteristic impedance [] • Z0,n : characteristic impedance of port n [] Radiofrequency Engineering C. Collado, J.M. González-Arbesú 5 EETAC-UPC INTRODUCTION Introduction • The low frequency analysis of circuits and networks (where dimensions are much lower than wavelength) can be analyzed by the already known Kirchhoff laws of voltage and current together with the impedance concepts of circuit theory. • High frequency regime requires different tools to account for the phase difference existing between one point and another in the circuits and networks. • Usually, the knowledge of the electromagnetics fields and/or currents and voltages in all points inside the network are not required. It uses to be enough knowing voltages or currents at a set of terminals. • A set of parameters will be defined in order to symplify the modelling of networks (or systems) accurately representing its performance. microstrip black layout box R R Parameters R /4 Radiofrequency Engineering C. Collado, J.M. González-Arbesú 6 EETAC-UPC INTRODUCTION Introduction • This model is useful for analyzing modifications of the network (design), combination of networks (at system or subsystem level), or analyzing the performance of the network when combined with its neighbors. • One has to be careful to not oversimplify the model of the circuit because it will lead to erroneous results. black box Parameters Radiofrequency Engineering C. Collado, J.M. González-Arbesú 7 EETAC-UPC INTRODUCTION Introduction • Let’s consider a microwave network (or system) having N terminals, also called ports. • The parameters modelling the network are usually defined at the ports. • These ports can be the physical ports of the network. However if a physical port supports more than one propagating mode, additional ports can be added to account for these modes. • Each port has a specific point called reference plane where parameters are measured. • The reference planes provide phase references for the voltage and current phasors. V , I t : reference plane; 3 3 V , I n N N V+ and I+: incident voltage and current waves; - - V3 , I3 V and I : reflected voltage VN , I N and current waves; V and I: total voltage and t3 current waves: tN Vn Vn Vn V1 , I1 I n I n I n V2 , I 2 V1 , I1 t t2 1 V2 , I2 Radiofrequency Engineering C. Collado, J.M. González-Arbesú 8 EETAC-UPC Z-PARAMETERS Z-Parameters • The impedance matrix [Z] relates the currents and the voltages in the terminals of the network: V Z I V1 Z11 Z12 ... Z1N I1 total total V2 Z21 Z22 ... I2 voltage at current at each port ... ... ... ... each port VN Z N1 ... ... Z NN I N transfer port impedances impedances . Zij may be complex. 2N2 degrees of freedom for an arbitrary network. For a reciprocal network (not containing any nonreciprocal media such as ferrites t or plasmas, or active devices), the [Z] matrix is symmetric: Zij = Zji ([Z]=[Z] ). If the network is lossless, the Zij elements are purely imaginary. Radiofrequency Engineering C. Collado, J.M. González-Arbesú 9 EETAC-UPC Z-PARAMETERS Z-Parameters • The total voltage at terminal i is: Vi Zi1I1 Zi2 I 2 ... ZiN I N i 1,..., N Vi • Each impedance in the [Z] matrix can be found as: Zij I j Ik 0 k j • To find the impedances of the [Z] matrix port j has to be driven with the current Ij leaving all other ports open-circuited. • A two-port network can be characterized by a 2x2 impedance matrix: I1 I 2 V1 Z11I1 Z12 I 2 + + V2 Z21I1 Z22 I 2 V1 [Z] V2 - - V1 V1 V2 V2 Z11 Z12 Z21 Z22 I1 I 2 I1 I 2 I2 0 I1 0 I2 0 I1 0 Radiofrequency Engineering C. Collado, J.M. González-Arbesú 10 EETAC-UPC Z-PARAMETERS Z-Parameters Example: Lossy network. Find the impedance matrix of the lossy network of the figure. R1 R2 I1 I 2 + + V1 R3 V2 - - V1 V1 Z11 R1 R3 Z12 R3 I1 I 2 I2 0 I1 0 R1 R3 R3 Z R3 R2 R3 V2 V2 Z21 R3 Z22 R2 R3 I1 I 2 I2 0 I1 0 Radiofrequency Engineering C. Collado, J.M. González-Arbesú 11 EETAC-UPC Y-PARAMETERS Y-Parameters • The admitance matrix [Y] relates the voltages and the currents in the terminals of the network: I Y V I1 Y11 Y12 ... Y1N V1 total total I2 Y21 Y22 ... V2 current at voltage at each port ... ... ... ... each port I N YN1 ... ... YNN VN transfer port admitances admitances . Yij may be complex. 2N2 degrees of freedom for an arbitrary network. For a reciprocal network (not containing any nonreciprocal media such as ferrites t or plasmas, or active devices), the [Y] matrix is symmetric: Yij = Yji ([Y]=[Y] ). If the network is lossless, the Yij elements are purely imaginary. Radiofrequency Engineering C. Collado, J.M. González-Arbesú 12 EETAC-UPC Y-PARAMETERS Y-Parameters • The total current at terminal i is: Ii Yi1V1 Yi2V2 ...YiNVN i 1,..., N Ii • Each impedance in the [Y] matrix can be found as: Yij V j Vk 0 k j • To find the impedances of the [Y] matrix port j has to be driven with the voltage Vj leaving all other ports short-circuited. • Impedance and admitance matrices are the inverses of each other: Y Z 1 • A two-port network can also be characterized by a 2x2 impedance matrix: I1 I 2 I1 Y11V1 Y12V2 + + I 2 Y21V1 Y22V2 V1 [Y] V2 - - I1 I1 I 2 I 2 Y11 Y12 Y21 Y22 V1 V2 V1 V2 V2 0 V1 0 V2 0 V1 0 Radiofrequency Engineering C. Collado, J.M. González-Arbesú 13 EETAC-UPC Y-PARAMETERS Y-Parameters Example: Lossy network. According to the definition, find the admitance matrix of the lossy network of the figure. Assess the result by applying the fact that the Y-matrix is the inverse of the Z-matrix. R1 R2 I1 I 2 + + V1 R3 V2 - - I1 1 R2 R3 Y11 V1 R1 R2 // R3 R1R2 R1R3 R2 R3 V2 0 I2 1 R1 R3 Y22 R2 R3 R3 V2 R2 R1 // R3 R1R2 R2 R3 R1R3 V1 0 R R R R R R R R R R R R Y 1 2 1 3 2 3 1 2 2 3 1 3 R R R I I R R R R 3 1 3 Y 2 1 3 2 3 3 21 R1R2 R1R3 R2 R3 R1R2 R2 R3 R1R3 V1 I1R1 R2 // R3 R1R2 R1R3 R2 R3 V2 0 I1 I2 R3 R1 R3 R3 Y12 V2 I2 R2 R1 // R3 R1R2 R2 R3 R1R3 V1 0 Radiofrequency Engineering C.