1. INTRODUCTION TO DISTRIBUTED CIRCUIT DESIGN . Microwave circuit elements and analysis • Frequency bands [COLLIN 1.1] • RF circuit analysis [COLLIN 1.3] . Transmission lines • Transmission line types • Propagation equations [COLLIN 3.1] [POZAR 3.1] forward and reverse o lossless lines propagating waves o lossy lines attenuation o low-loss lines • Reflection coefficient [POZAR 3.3] impedance of transmission lines • Power and losses: return loss [POZAR 3.3] • Voltage standing wave ratio [POZAR 3.3] • Impedance [POZAR 3.3] • Generator mismatch [POZAR 3.6] • Smith Chart [COLLIN 5.1] • Impedance matching [COLLIN 5.2-5.5] o lumped elements o Single-stub matching o Double-stub matching o Quarter-wave impedance transformer Radiofrequency Engineering C. Collado, J.M. González-Arbesú 1 EETAC-UPC 1. INTRODUCTION TO DISTRIBUTED CIRCUIT DESIGN . Transmission line design • Balanced and unbalanced lines • Homogeneous and non-homogeneous lines • Coupled lines [POZAR 8.6] • Line design . Application notes • Coaxial cables • Connectors [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 • : attenuation constant [m-1] • : phase constant [rad·m-1] • Cd : distributed capacitance per unit length [F/m] -12 • 0 : electric permittivity of vacuum [8.85·10 F/m] • f0 : frequency [Hz] • : propagation constant [m-1] • G : distributed conductance per unit length [S/m] • i(z,t) : current in time domain [V] + • I0 : current amplitude of progressive wave at z=0 [A] • l : transmission line length [m] • Ld : distributed inductance per unit length [H/m] • : wavelength [m] -7 • 0 : magnetic permeability of vacuum [4·10 H/m] • : angular frequency [rad/s] • R : distributed resistance per unit length [/m] • T : period [s] Radiofrequency Engineering C. Collado, J.M. González-Arbesú 3 EETAC-UPC GLOSSARY • RS : surface resistivity [/square] • S : skin depth [m] • : conductivity [Sm] • d : dielectric conductivity [Sm] • tan : loss tangent [adim] • RL : return loss [dB] • : (voltage) reflection coefficient [adim] • G : (voltage) generator reflection coefficient [adim] • IN : (voltage) reflection coefficient at input port [adim] • L : (voltage) load reflection coefficient [adim] • v(z,t) : voltage in time domain [V] • VG : voltage at generator [V] + • V0 : voltage amplitude of progressive wave at z=0 [V] • vp : phase velocity [m/s] • VSWR : Voltage Standing Wave Ratio [adim] • ZG : generator impedance [] • ZIN : impedance at the input port of the transmission line [] • ZL : load impedance [] • Z0 : transmission line characteristic impedance [] Radiofrequency Engineering C. Collado, J.M. González-Arbesú 4 EETAC-UPC MICROWAVE CIRCUIT ELEMENTS AND ANALYSIS Frequency bands • International classification of the frequency bands. • Radar classification of frequency bands. The old one (WW II) is still widely used. [Tables taken from: R.E. Collin, Foundations for Microwave Engineering, Wiley-Interscience, 2nd Edition, 2001 (New York)] Radiofrequency Engineering C. Collado, J.M. González-Arbesú 5 EETAC-UPC MICROWAVE CIRCUIT ELEMENTS AND ANALYSIS RF circuit analysis • At frequencies where is several orders of magnitude larger than the greatest dimension of the circuit or system: . To transmit, receive, and/or process data the basic building blocks are capacitors, inductors, resistors, and transistors. Loop currents and node voltages are enough to analyse the circuits. To analyse the circuits no propagation effects have to be considered: the delay in the propagation of signals at different points in the circuit is negligible compared with the period of the applied signal. Lumped circuit models are valid. • At microwave frequencies is compared with the circuit dimensions and: . Propagation effects can not be ignored: there is a delay in the propagation of signals among different points in a circuit. There are distributed capacitances and inductances in the circuit. There is an increase in the impedance of terminals and connectors. Unshielded circuits with dimensions compared with become effective radiators. Distributed circuit models are used. Radiofrequency Engineering C. Collado, J.M. González-Arbesú 6 EETAC-UPC MICROWAVE CIRCUIT ELEMENTS AND ANALYSIS RF circuit analysis • Wave propagation along a line considering the propagation delays: L: line length t = 0 Source voltage: v: propagation speed vG t A cos0t propagation delay v (t) G Voltage: z vz,t A cos0 t v z = 0 z • Some remarks: . Wave propagation along a line considering the propagation delays. Each point in the line has a different voltage/current at the same time t. Periodicity in time or period T. 2 2 v T . Spatial periodicity or wavelength . 0 0 v f0 . Dimensions use to be defined with respect to . Radiofrequency Engineering C. Collado, J.M. González-Arbesú 7 EETAC-UPC MICROWAVE CIRCUIT ELEMENTS AND ANALYSIS RF circuit analysis Example: Propagation at Low Frequencies. Consider a circuit having a transmission line length of 0.003 (3 lines, 0.001-length each) fed with a sinusoidal wave of 1 GHz. V1 V2 V3 V4 VtSine R TRANSIENT SRC1 R TLIN TLIN TLIN R2 Vdc=0 V R1 TL1 TL2 TL3 R=50 Ohm Tran Amplitude=1 V R=50 Ohm Z=50.0 Ohm Z=50.0 Ohm Z=50.0 Ohm Tran1 Freq=1 GHz E=0.36 E=0.36 E=0.36 StopTime=10.0 nsec Delay=0 nsec F=1 GHz F=1 GHz F=1 GHz MaxTimeStep=1.0 psec 600 600 600 600 400 400 400 400 200 200 200 200 0 0 0 0 V4, mV V1, mV -200 V2, mV -200 V3, mV -200 -200 -400 -400 -400 -400 -600 -600 -600 -600 01024 6 8 0102468 0102468 0102468 time, nsec time, nsec time, nsec time, nsec Radiofrequency Engineering C. Collado, J.M. González-Arbesú 8 EETAC-UPC MICROWAVE CIRCUIT ELEMENTS AND ANALYSIS RF circuit analysis Example: Propagation at High Frequencies. Consider a circuit having a transmission line length of 15 (3 lines, 5-length each) fed with a sinusoidal wave of 1 GHz. V1 V2 V3 V4 VtSine R TRANSIENT SRC1 R TLIN TLIN TLIN R2 Vdc=0 V R1 TL1 TL2 TL3 R=50 Ohm Tran Amplitude=1 V R=50 Ohm Z=50.0 Ohm Z=50.0 Ohm Z=50.0 Ohm Tran1 Freq=1 GHz E=1800 E=1800 E=1800 StopTime=10.0 nsec Delay=0 nsec F=1 GHz F=1 GHz F=1 GHz MaxTimeStep=1.0 psec 600 600 600 600 400 400 400 400 200 200 200 200 0 0 0 0 V4, mV V1, mV V1, -200 mV V2, -200 mV V3, -200 -200 -400 -400 -400 -400 -600 -600 -600 -600 01024 6 8 01024 6 8 01024 6 8 01024 6 8 time, nsec time, nsec time, nsec time, nsec Radiofrequency Engineering C. Collado, J.M. González-Arbesú 9 EETAC-UPC TRANSMISSION LINES Transmission line types • Transmission lines are physical devices whose purpose is to guide electromagnetic waves (carry RF power) from one place to another. • They are capable of guiding TEM waves (TEM waves can only exist in structures containing two or more separated conductors). • Two-wire transmission lines are inefficient for transfering electromagnetic energy at high frequencies due to the lack of confinement in all directions. • Coaxials are more efficient than two-wire lines in those cases. two-wire ribbon line twisted pair (twin lead) shielded pair air coaxial line flexible coaxial line [Images from: http://www.techlearner.com/Apps/TransandGuides.pdf] Radiofrequency Engineering C. Collado, J.M. González-Arbesú 10 EETAC-UPC TRANSMISSION LINES Transmission line types • There are lots of planar structures used as transmission lines. Metallic parts are supported by dielectrics (fiberglass, ceramics, foams,...). microstrip coplanar transmission line stripline Radiofrequency Engineering C. Collado, J.M. González-Arbesú 11 EETAC-UPC TRANSMISSION LINES Transmission line types • Waveguides are the most efficient. They are fabricated with just one conductor. Waveguides do not support TEM waves. • Two-wire lines are less bulky and less expensive than waveguides. waveguides [Image from: http://www.techlearner.com/Apps/TransandGuides.pdf] Radiofrequency Engineering C. Collado, J.M. González-Arbesú 12 EETAC-UPC TRANSMISSION LINES Propagation equations • Field modes: electromagnetic fields configurations supported by a structure. • A coaxial transmission line supports a TEM mode (electric field orientation, magnetic field orientation, and energy propagation direction for a triad). z z H + iz,t L vz,t b E - L a abstract model (ideal transmission line) electromagnetic field distribution physical structure vz,t Ez,t dl (TEM mode) iz,t H z,t dl Radiofrequency Engineering C. Collado, J.M. González-Arbesú 13 EETAC-UPC TRANSMISSION LINES Propagation equations: lossless lines • The knowledge of voltage and current waves propagating along the transmission line allows the use of a distributed circuit model to analyze its performance. • The model represents an infinitesimally short segment of the transmission line • This model is convenient to explore properties of lines without knowing the fields in detail. However, the structures should be analyzed in detail if accurate performances have to be known. iz,t iz z,t iz,t iz z,t + + + L + vz,t vz z,t vz,t vz z,t C - - - - L L z z d or dz in case that z 0 C Cd z • Ld and Cd are the distributed inductance [H/m] and capacitance [F/m] associated to the coaxial structure and materials. No losses are assumed in this example (meaning that there is no distributed resistance). Radiofrequency Engineering C. Collado, J.M. González-Arbesú 14 EETAC-UPC TRANSMISSION LINES Propagation equations: lossless lines • Applying Kirchhoff’s voltage and current laws: iz,t iz z,t iz,t vz,t Ld z vz z,t + + t L vz,t vz z,t vz z,t C iz,t C z iz z,t - - d t L Ld z C Cd z • Dividing by z and taking • Considering sinusoidal steady-state condition the limit z 0: (cosine based phasors) (TRANSIENTS NOT vz,t iz,t CONSIDERED): V L j L I z d t z d iz,t vz,t I C j C V z d t z d Radiofrequency Engineering C.