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Microwave Engineering & Special Topics in Passive Microwave Circuit Design Dept. Electrical and Information Engineering SeoulTech Jae-Young Chung...

Microwave Engineering & Special Topics in Passive Microwave Circuit Design Dept. Electrical and Information Engineering SeoulTech Jae-Young Chung 1 Electromagnetic Measurement & Application (EMMA) Lab  EMMA Lab. https://sites.google.com/site/jychung2012/ 2 Electromagnetic Measurement & Application (EMMA) Lab  eclass http://eclass.seoultech.ac.kr/ Please make sure you have an access for eclass. If not, ask your department admin or advisor. 3 Electromagnetic Measurement & Application (EMMA) Lab  eclass Senior undergraduates + Graduate students (Graded separately) In-class (off-line) lecture on Monday Video (on-line) lecture for Friday Lecture materials (ppt slides) will be uploaded every Sunday. On-line materials (videos) will be uploaded every Thursday. Please use Q&A bulletin if you have a question related to the class or lecture. Otherwise, send me an email. ([email protected]) Do not send a message via eclass. 4 Electromagnetic Measurement & Application (EMMA) Lab  MWE Book 2nd ed. 3rd ed. 4th ed. Any edition will be fine. 5 Electromagnetic Measurement & Application (EMMA) Lab  Homework 0.8 QWT BMT 0.6 CMT | | 0.4 0.2 0 0 500 1000 1500 2000 Frequency (MHz) 6 Electromagnetic Measurement & Application (EMMA) Lab  Calendar 9월 10월 11월 12월 I hope we have a great semester together! 7 Electromagnetic Measurement & Application (EMMA) Lab  What is Microwave? 1 Electromagnetic Measurement & Application (EMMA) Lab  Frequency Spectrum c Speed of light = f wavelength frequency 2 Electromagnetic Measurement & Application (EMMA) Lab  Terminology Wireless (Antenna) RF cable Optical cable VLF: Very Low Frequency (3 kHz – 30 kHz) Radio Frequency (3 kHz – 300 GHz) LF: Low Frequency (30 kHz – 300 kHz) Microwave (300MHz – 300 GHz) MF: Medium Frequency (300 kHz – 3 MHz) Millimeter-wave (30 GHz – 300 GHz) HF: High Frequency (3 MHz – 30 MHz) Submillimeter-wave (300 GHz – 3 THz) VHF: Very High Frequency (30 MHz – 300 MHz) Terahertz wave (0.3 THz – 3 THz) UHF: Ultra High Frequency (300 MHz – 3 GHz) SHF: Super High Frequency (3 GHz – 300 GHz) EHF: Extremely High Frequency (30 GHz – 300 GHz) L-band (1-2 GHz), S-band (2-4 GHz), C (4-8 GHz), X (8-12 GHz), Ku (12-18 GHz), K (18-26 GHz), Ka (26-40 GHz), V (40-75 GHz), W (75-110 GHz), D (110-170 GHz) 3 Electromagnetic Measurement & Application (EMMA) Lab  Wireless Communications https://terasense.com/terahertz-technology/radio-frequency-bands/ 4 Electromagnetic Measurement & Application (EMMA) Lab  Radars Body scanner Automotive Surface movement Airborne Precision approach Battle field Airport surveillance Early warning En-route GPR https://www.radartutorial.eu/07.waves/Wave s%20and%20Frequency%20Ranges.en.html Over-The-Horizon 5 Electromagnetic Measurement & Application (EMMA) Lab  Frequency Spectrum, Korea https://spectrummap.kr/ 6 Electromagnetic Measurement & Application (EMMA) Lab  Frequency Spectrum, Cellular Service 7 Electromagnetic Measurement & Application (EMMA) Lab  5G (Sub-6GHz) 8 Electromagnetic Measurement & Application (EMMA) Lab  5G (mmWave) 9 Electromagnetic Measurement & Application (EMMA) Lab  5G (mmWave) 10 Electromagnetic Measurement & Application (EMMA) Lab  Frequency Band: Continue to Increase Early 20th: up to few MHz → AM radio (~1.6 MHz) Mid 20th: up to 500 MHz→ FM radio (~100MHz), UHF-TV (~470MHz) Late 20th: up to 1 GHz → Cell phone (CDMA, GSM ~ 900 MHz) Early 21th : up to few GHz → 4G (LTE ~2.3GHz), Bluetooth (2.4GHz), WiFi (2.4, 5.8 GHz) Now: 5G (3.5GHz, 28GHz, 40GHz), Wifi 6 (5.9~7.1GHz), UWB (6.5~9GHz), LEO (Ku/Ka) Why?? → More bandwidth (more data, higher resolution, faster clock speed) RF components (including antenna) are smaller More antenna gain in the same footprint Promote small scale analysis (bio-engineering, nano-engineering) 11 Electromagnetic Measurement & Application (EMMA) Lab  Blueprint for 6G https://orbray.com/magazine_en/archives/385 12 Electromagnetic Measurement & Application (EMMA) Lab  Sub-THz for 6G W-band, D-band or even higher G-band, H-band 13 Electromagnetic Measurement & Application (EMMA) Lab  The Problem: Attenuation & Blockage Attenuation & blockage is significant at mmWave & sub-THz wave frequencies c = f http://web.tiscali.it/ Also, components are more vulnerable to loss, coupling, and parasitic effects causing non-linear behavior → Components are small but expensive 14 Electromagnetic Measurement & Application (EMMA) Lab  Example: Nonlinear Behavior Keysight, Application note, “Understanding the Fundamental Principles of Vector Network Analysis” 15 Electromagnetic Measurement & Application (EMMA) Lab  Mid-Band for 6G → Centimeter wave frequency is promising https://www.nokia.com/about-us/newsroom/articles/spectrum-for-6G-explained/ https://www.ericsson.com/en/blog/2022/6/6g-spectrum-why-its-fundamental 16 Electromagnetic Measurement & Application (EMMA) Lab  What is Microwave ? What is Microwave Engineering ? (RF Engineering) 1 Electromagnetic Measurement & Application (EMMA) Lab  Wireless “Network” Top-down approach 2 Electromagnetic Measurement & Application (EMMA) Lab  Wireless “Service” Wireless communication technologies, protocols 3 Electromagnetic Measurement & Application (EMMA) Lab  Wireless “Device” CPU/GPU PMIC Display Multiplexer FEM FEM PMIC PMIC PMIC UWB Module Display Power Modem Printed Circuit Board (PCB) Surface Mount Technology (SMT) RF Xcvr Wi-Fi/BT Module PMIC MCU/eSIM Envelope Tracker Envelope Tracker FEM Wireless Charging IC https://www.techinsights.com/blog/teardown/apple-iphone-13-pro-teardown 4 Electromagnetic Measurement & Application (EMMA) Lab  RF Front-End “Module” Introduction to VNA Basics, Tektronix, Technical Notes 5 Electromagnetic Measurement & Application (EMMA) Lab  RF “Component” Integration Antennas Component type Network analyzer basics, Agilent, technical note 6 Electromagnetic Measurement & Application (EMMA) Lab  Active vs. Passive Component Active Passive Do need an external source Do not need external source Deliver/Produce energy to circuit Utilize/Store energy from circuit Can control the flow of current Cannot control the flow of current www.codrey.com 7 Electromagnetic Measurement & Application (EMMA) Lab  Passive RF Component We are going to learn basic design theories of passive RF components such as transmission lines, matching circuits, power dividers, couplers, isolators, filters, etc. https://www.minicircuits.com/ 8 Electromagnetic Measurement & Application (EMMA) Lab  Planar RF Circuit Printed Circuit Board (PCB): A cheap way to implement and mass-product a circuit 9 Electromagnetic Measurement & Application (EMMA) Lab  EM Fundamentals 1 Electromagnetic Measurement & Application (EMMA) Lab  Electromagnetic Wave Assuming you have attended 1 or 2 electromagnetic classes Static electric field Electromagnetic field Electromagnetic wave Static magnetic field Time-varying Propagate, Radiate 2 Electromagnetic Measurement & Application (EMMA) Lab  Guided & Radiated Waves Waveguide Coaxial cable Microstrip line E H Dipole antenna Phased array https://www.emcos.com/ 3 Electromagnetic Measurement & Application (EMMA) Lab  Maxwell’s Equation Google images 4 Electromagnetic Measurement & Application (EMMA) Lab  Maxwell’s Equations Q  S E  ds =    D = v (Gauss’s Law) d B B  E  dl = − C dt E = − t (Faraday’s Law)  B  ds = 0 S B = 0 (Magnetic Gauss’s Law) D D  C H  dl = I +  S t  ds H = J + t (Ampere-Maxwell Law) 5 Electromagnetic Measurement & Application (EMMA) Lab  Gauss’s Law The total flux of field lines penetrating the surface is the same and depends only on the amount of charge inside. Q   D = v  S E  ds =  Magnetic Gauss’s law: Divergence of a magnetic field is zero → sum of going in and out B are the same.  B  ds = 0 S B = 0 6 Electromagnetic Measurement & Application (EMMA) Lab  Ampere-Maxwell’s Law Ampere’s law: the circulation of the magnetic flux density in a medium around any closed path is equal to the total current flowing through the surface bounded by the path. C H  dl = I H = J l Ampere-Maxwell’s law: Time-varying electric flux also generates magnetic fields. D D  C H  dl = I +  S t  ds H = J + t 7 Electromagnetic Measurement & Application (EMMA) Lab  Faraday’s Law of Induction The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux enclosed by the circuit. → Time-varying magnetic flux generates electric fields d B B CE  dl = − dt E = − t 8 Electromagnetic Measurement & Application (EMMA) Lab  Maxwell’s Eq. in Phasor Form All of the fields, currents, and voltages are considered to be sinusoidal of radian frequency 𝝎 (unless noted) → Time-Harmonic Phasor = A complex number which holds the magnitude and phase of a sinusoid v(t ) = Acos(ω t + θ ) Omit time dependence V = A = Ae j = A cos  + jA sin  Phasor E ( x, y , z ; t ) E ( x, y, z ) = Ex xˆ + E y yˆ + Ez zˆ = Re E ( x, y, z ) e j t  Ex ( x, y, z ) = Ex  x Maxwell’s equation in phasor form  E(x, y, z; t ) → j E(x, y, z ) t 9 Electromagnetic Measurement & Application (EMMA) Lab  Electromagnetic Material Properties Material properties that affect electric and magnetic fields in a macroscopic viewpoint. Permittivity → How well is a dielectric polarized?  Permeability → How well is a magnetic material magnetized?  Conductivity → How well is a conductor free from loss(resistivity)?  D=E D : Electric flux density (Coul/m2) E : E-field intensity (V/m) Constitutive relations B = H B : Magnetic flux density (Wb/m2) H : H-field intensity (A/m) J = E J : Electric current density (A/m2)  =  0 r [F/m] 1 = [S/m]  = 0 r [H/m]  1 Electromagnetic Measurement & Application (EMMA) Lab  Microscopic View in Materials Dielectric Random Polarized Random Magnetized Magnetic material High  (or ) → easily polarized (or magnetized) → permit more electric (or magnetic) field → store more electric (or magnetic) energy 2 Electromagnetic Measurement & Application (EMMA) Lab  Types of Material Isotropic vs. anisotropic (orientation dependent)  xx  xy  xz    xx  xy  xz  y      =  0  yx  yy  yz     =  0   yx  yy  yz  x  zx  zy  zz    zx  zy  zz  z   Homogeneous vs. inhomogeneous (position dependent)   ( x, y , z )   ( x, y , z ) ※ Simple medium - isotropic Linear vs. non-linear (field strength dependent) - homogeneous  (H ) - linear   (E )  - non-dispersive → ,  are real constants Non-dispersive vs. dispersive (frequency dependent)  (f )  ( f ) Electromagnetic Measurement & Application (EMMA) Lab  Dispersive material Permittivity (or permeability) is a function of frequency (frequency dependent). → Permittivity (or permeability) is a complex number. Real part → how much energy is stored Imag. part → how much energy is dissipated (loss) Complex permittivity Relative permittivity Permeability Relative permeability   =  0 r =  0 (  r − j r)   = 0 r = 0 ( r − j r) Permittivity of air = 8.854 x 10-12 (F/m) Permeability of air = 4p x 10-7 (H/m) Dk & Df Relative permittivity (Dielectric constant, Dk)     r    =  0 r =  0 (  r − j r) =  0 r (1 − j tan  ) =  0   r − j  tan  = =   0   r  0 r Loss tangent (Df) 4 Electromagnetic Measurement & Application (EMMA) Lab  Frequency Spectrum Dipolar  relaxation Dielectrics Atomic polarization Electronic polarization   f (Hz) Magnetic materials Domain wall resonance Ferromagnetic resonance 5 Electromagnetic Measurement & Application (EMMA) Lab  Chart, Permittivity Material Dielectric Constant (relative to air) Loss tangent ABS (plastic), Molded 2.0 - 3.5 0.00500 - 0.0190 Air 1.00054 Alumina - 96% 10.0 0.0002 @ 1 GHz - 99.5% 9.6 0.0002 @ 100 MHz 0.0003 @ 10 GHz 2.35 @ 1 MHz 0.001 @ 1 MHz Butyl Rubber 2.35 @ 3 GHz 0.002 @ 3 GHz FR-4 (G-10) - low resin 4.9 0.008 @ 100 MHz - high resin 4.2 0.01 @ 3 GHz 0.0002 @ 100 MHz Fused quartz 3.8 0.0004 @ 3 GHz Gallium Arsenide (GaAs) 13.1 0.0016 @ 10 GHz Glass (Corning 7059) 5.75 0.0036 @ 10 GHz 4.15 @ 1 MHz 0.12 @ 1 MHz Ice (pure distilled water) 3.2 @ 3 GHz 0.0009 @ 3 GHz 2.25 @ 1 MHz 0.025 @ 1 MHz Mahogany 1.88 @ 3 GHz 0.025 @ 3 GHz 0.0001 @ 100 MHz Polystyrene 2.5 - 2.6 0.00033 @ 3 GHz 0.005 @ 1 GHz Silicon 11.7 - 12.9 0.015 @ 10 GHz 2.59 @ 1 MHz 0.017 @ 1 MHz Soil (dry sandy) 2.55 @ 3 GHz 0.0062 @ 3 GHz Teflon® (PTFE) 2.0 - 2.1 0.00028 @ 3 GHz Vacuum (free space) 1.00000 Water (32°F) 88.0 0.04 @ 1 MHz (68°F) 80.4 0.157 @ 3 GHz (212°F) 55.3 0.04 @ 0.1 GHz 6 Wood 1.2 - 2.1 0.03 @ 3 GHz Electromagnetic Measurement & Application (EMMA) Lab  Chart, Conductivity Material ρ (Ω·m) at 20 °C σ (S/m) at 20 °C Carbon (graphene) 1.00×10−8 1.00×108 Silver 1.59×10−8 6.30×107 Copper 1.68×10−8 5.96×107 Gold 2.44×10−8 4.10×107 Aluminum 2.82×10−8 3.50×107 Tungsten 5.60×10−8 1.79×107 Zinc 5.90×10−8 1.69×107 Nickel 6.99×10−8 1.43×107 Lithium 9.28×10−8 1.08×107 Iron 1.00×10−7 1.00×107 Platinum 1.06×10−7 9.43×106 Tin 1.09×10−7 9.17×106 Carbon steel 1.43×10−7 6.99×106 Lead 2.20×10−7 4.55×106 Titanium 4.20×10−7 2.38×106 Stainless steel 6.90×10−7 1.45×106 Mercury 9.80×10−7 1.02×106 GaAs 1.00×10−3 to 1.00×108 1.00×10−8 to 103 Sea water 2.00×10−1 4.80 Silicon 6.40×102 1.56×10−3 Wood (damp) 1.00×103 to 1.00×104 10−4 to 10−3 Glass 1.00×1011 to 1.00×1015 10−15 to 10−11 Hard rubber 1.00×1013 10−14 Air 1.30×1016 to 3.30×1016 3×10−15 to 8×10−15 PET 1.00×1021 10−21 Teflon 1.00×1023 to 1.00×1025 10−25 to 10−23 7 Electromagnetic Measurement & Application (EMMA) Lab  Wave Equation & Plane Wave 1 Electromagnetic Measurement & Application (EMMA) Lab  Maxwell’s Eq. (Recap.) Q  E  ds = S 0   D = v (Gauss’s Law) d B B CE  dl = − dt E = − t (Faraday’s Law)  B  ds = 0 S B = 0 (Magnetic Gauss’s Law) D D  C H  d l = I +  t  ds S H = J + t (Ampere-Maxwell Law) Maxwell’s equation in phasor form  E(x, y, z; t ) → j E(x, y, z ) t 2 Electromagnetic Measurement & Application (EMMA) Lab  Wave Equation Phasor form of Maxwell’s equation   E = − j H Source-free   H = j E Time harmonic J = 0, v = 0 E = 0 H = 0 Curl left and right sides E and H are  E +   E = 0 2 2 uncoupled     E = − j  H =  2  E  2 H +  2  H = 0     E =  (   E ) −  2E k =   wave number  2E + k 2E = 0 Wave equation 2H + k 2H = 0 3 Electromagnetic Measurement & Application (EMMA) Lab  Plane Wave  2E + k 2E = 0 2H + k 2H = 0 We can solve for the wave equation to find E- or H-field. If you solve for E(or H), then H (or E) can be obtained from   E = − j H   H = j E Solutions of the wave equation are countless. A unique solution can be obtained if boundary conditions (B.C.) are provided. The simplest solution with the simplest B.C. (free-space) is “Plane wave” 4 Electromagnetic Measurement & Application (EMMA) Lab  Plane Wave Plane wave is a transverse electromagnetic (TEM) wave that is constant in both magnitude and direction, over a plane normal to the direction of propagation. (Poynting vector direction) P = E H - At a location, all the E and H-fields exist on a plane (The wavefront is plane). - No real plane wave exist. It’s a simplified idealization. - Nevertheless, it’s an effective approximation for many practical problems. 5 Electromagnetic Measurement & Application (EMMA) Lab  Plane Wave Solution & Parameters  2 E +  2  E = 0 E x ( z ) = xˆ Ex ( z ) = xˆ Ex ( z ) x ( z ) E x ( z , t ) = xˆ Ex ( z , t ) = xˆ E x cos ( t − kz ) Wavelength Angular frequency vp Amplitude  = 2f  up f = = 2  Frequency Period 2 T=    = = vp k  Wave number Intrinsic impedance Phase velocity (velocity of wave front) (Wave impedance) 6 Electromagnetic Measurement & Application (EMMA) Lab  Wave Impedance Impedance Wave impedance in air (vacuum) V E   0 Z= = R + jX    ZW =  = = = Z 0 = 0 = = 120  = 377  I H k  0 7 Electromagnetic Measurement & Application (EMMA) Lab  Example 8 Electromagnetic Measurement & Application (EMMA) Lab  Transmission line (tx-line, t-line) ZS Source Z0 ZL 1 Electromagnetic Measurement & Application (EMMA) Lab  Tx-Lines 2 Electromagnetic Measurement & Application (EMMA) Lab  Poynting Vector P = E H x z y H H Wire I P P E E Closed-loop (Circuit) P 3 en.wikipedia.org/wiki/Poynting_vector Electromagnetic Measurement & Application (EMMA) Lab  Poynting Vector Two conductor line Coaxial line P P P P F. Gustrau, RF and Microwave Engineering: Fundamentals of Wireless Communications, John Wiley & Sons, 2012. 4 Electromagnetic Measurement & Application (EMMA) Lab  Types of Tx-line Cable Waveguide Microstrip line Coplanar waveguide www.colemanmw.com/ 5 Electromagnetic Measurement & Application (EMMA) Lab  Microstrip Lines (Planar Tx-Lines) Stripline Microstrip line Coplanar waveguide Slot waveguide Substrate Integrated Waveguide (SIW) en.wikipedia.org/wiki/Planar_transmission_line 6 Electromagnetic Measurement & Application (EMMA) Lab  Microwave Circuits Circuits based on tx-line elements 7 Electromagnetic Measurement & Application (EMMA) Lab  Tx-Line Comparison Table D.M. Pozar, Microwave Engineering, 4th ed., Wiley, 2011 8 Electromagnetic Measurement & Application (EMMA) Lab  TEM, TE, TM Waves Transverse Electric and Magnetic Wave (TEM) : Neither E- nor H-field in the direction of propagation : Both E- and H-fields are orthogonal to the direction of propagation Transverse Electric Wave (TE, H-mode) : No E-field in the direction of propagation : Only E-field is orthogonal to the direction of propagation Transverse Magnetic Wave (TM, E-mode) : No H-field in the direction of propagation : Only H-field is orthogonal to the direction of propagation http://sub.allaboutcircuits.com 9 Electromagnetic Measurement & Application (EMMA) Lab  Waveguide Modes and Cutoff Frequency Cutoff frequency 2 2 1 m n f0 =   +  2   a  b 10 Electromagnetic Measurement & Application (EMMA) Lab  Rect. Waveguide Chart https://en.wikipedia.org/wiki/Waveguide_(electromagnetism) 11 Electromagnetic Measurement & Application (EMMA) Lab  Notes Tx-line is used for efficiently transmitting electromagnetic energy over a distance. Electromagnetic waves generated by a time-varying source are propagated through a tx-line by the interaction of time-varying E- and H-fields. The EM energy is delivered toward the Poynting vector direction. Plane wave can be used to analyze TEM waves guided by a tx-line.  2 E +  2  E = 0 E x ( z ) = xˆ Ex ( z ) = xˆ Ex ( z ) x ( z ) E x ( z , t ) = xˆ Ex ( z , t ) = xˆ E x cos ( t − kz ) vp 2 f 1 E   f = vp = = ZW =  = = =  k  H k  1 Electromagnetic Measurement & Application (EMMA) Lab  Tx-Line Analysis We often prefer circuit analysis instead of EM. Maxwell’s eq., wave eq., Kirchhoff, Thevenin, Norton, E- and H-fields, vectors, vs. V and I, signals, scalar, modes, polarization, components and tx-lines, boundary condition, ports, terminals, loops structure, geometry 2 Electromagnetic Measurement & Application (EMMA) Lab  Equivalent Circuit Model Complicated structure can be represented by Circuit symbols Complicated circuit can be represented by Thevnin (V), Norton (I) model We don’t need to know (don’t want to know) what’s happening inside 3 Electromagnetic Measurement & Application (EMMA) Lab  Equivalent Circuit Model Lumped model vs. distributed model l   Circuit is much smaller than wavelength Lumped model: Currents do not vary spatially l  Circuit is comparable or larger than wavelength Distributed model: Currents do vary spatially → Microwave tx-line model 4 Electromagnetic Measurement & Application (EMMA) Lab  Tx-Line Distributed Model Tx-line at microwave frequency → Use distributed circuit model Model for a unit length Distributed elements for a unit length R: resistance (Ω m ) L: inductance (H m ) C: capacitance (F m ) G: conductance ( S m ) 5 Electromagnetic Measurement & Application (EMMA) Lab  Tx-Line Distributed Model Conduction currents generate - Resistance, R, due to losses in the conductors ΦB - Inductance, L, due to magnetic flux linking the current path L= I Displacement currents generate Q C= - Conductance, G, due to losses in the dielectric V - Capacitance, C, due to potential difference between conductors 6 Electromagnetic Measurement & Application (EMMA) Lab  Tx-Line Distributed Model Voltage charges capacitance, current charges inductance, and so on. www.allaboutcircuits.com 7 Electromagnetic Measurement & Application (EMMA) Lab  Tx-line Parameters R, L, C, G can be calculated from field quantities and B.C. D.M. Pozar, Microwave Engineering, 4th ed., Wiley, 2011 8 Electromagnetic Measurement & Application (EMMA) Lab  Tx-line Calculator http://www.awrcorp.com/ 9 Electromagnetic Measurement & Application (EMMA) Lab  Tx-Line Circuit Analysis (KVL) Applying simple circuit analysis on a unit length tx-line di vL = L dt KVL on loop: Phasor form: V ( z ) = R z I ( z ) + jL z I ( z ) + V ( z + z ) V (z + z ) − V (z ) Divide by z: = −(R + jL )I (z ) z dV (z ) lim z → 0 Limit z→0: = −(R + jL )I (z ) (1) dz 1 Electromagnetic Measurement & Application (EMMA) Lab  Tx-Line Circuit Analysis (KCL) dv iC = C dt KCL on node: Phasor form: I ( z ) = GzV ( z + z ) + jCzV ( z + z ) + I ( z + z ) Divide by z: I (z + z ) − I (z ) = −(G + jC )V (z + z ) z dI (z ) lim z → 0 Limit z→0: = −(G + jC )V (z )  (2) dz 2 Electromagnetic Measurement & Application (EMMA) Lab  Telegrapher Equation dV (z ) = −(R + jL )I (z ) (1) dz Telegrapher equation dI (z ) = −(G + jC )V (z )  (2) dz Take derivative of (1) and plug into (2) d 2V (z ) dI ( z ) 2 = − ( R + j L ) = (R + jL )(G + jC )V (z ) dz dz = (R + j L )(G + j C ) =  + j Complex propagation constant Attenuation Phase constant constant Similarly, take derivative of (2) and plug into (1) 3 Electromagnetic Measurement & Application (EMMA) Lab  Tx-Line Wave Equation Spatial distribution of V and I along the transmission line (1-D 2nd order differential equation)  2E + k 2E = 0 2H + k 2H = 0 cf. 1-D wave equation d 2 E x (z ) 2 d 2 H y (z ) 2 + k E x (z ) = 0 + k 2 H y (z ) = 0 dz dz 2 Tx-line’s V and I follow wave equation → Circuit where V and I have wave-like behaviors 4 Electromagnetic Measurement & Application (EMMA) Lab  Solution of Tx-Line Wave Equation Solution is the addition of +z forward and –z backward waves. V ( z , t ) = Re Vo+ e − ( + j  ) z + Vo− e( + j  ) z  e jt  j ( t −  z ) j ( t +  z ) = Re Vo+ e − z e + Vo− e z e   = (R + j L )(G + j C ) =  + j = Vo+ e − z cos (t −  z ) + Vo− e z cos (t +  z ) V0+ e − z V0− e z are determined by B.C. 5 Electromagnetic Measurement & Application (EMMA) Lab  Characteristic Impedance can be expressed in terms of dV (z ) = −(R + j L ) I (z )(1) dz dV ( z ) I (z ) = − 1 R + j L dz =− 1 (Vo e + Vo− e + z ) d + − z R + j L dz   Vo+  − z  − Vo−  +  z =− 1 R + j L ( − Vo e +  Vo e ) =  R + j L  e +  R + j L  e + − z − + z     = I o+ e − z + I o− e +  z Characteristic impedance of a tx-line Electromagnetic Measurement & Application (EMMA) Lab  Propagation Constant 2  = (R + jL )(G + jC ) =  + j =  = vp attenuation constant phase constant e − z = e −( + j ) z = e − z e − j z Determine phase Determine magnitude e −  L = e − L e − j  L 7 Electromagnetic Measurement & Application (EMMA) Lab  Summary V and I along a tx-line length direction (z-direction) are Combination of forward and backward waves V0+ e − z − z Z0 V e 0 2 The waves guided by a tx-line has a wavelength of  =   propagates with the velocity of v p =  attenuates by e − z The characteristic of a tx-line is represented by the characteristic impedance 8 Electromagnetic Measurement & Application (EMMA) Lab  Lossy Tx-Line Complex permittivity   =  0 (  r − j r) =  0 r (1 − j tan  )   tan  =  →R  r Complex propagation constant   → G = ( R + j L ) ( G + jC ) =  + j  Attenuation constant Phase constant Complex characteristic impedance R + j L Z0 = = Re ( Z 0 ) + j Im ( Z 0 ) G + jC → Lossy tx-line’s ( Z0 ,  ) are dispersive. (frequency dependent) 1 Electromagnetic Measurement & Application (EMMA) Lab  Lossless Tx-Line Typically, a tx-line is designed for efficient EM energy transfer. → Loss is small thus can be neglected in many practical cases. purely imag. = (R + jL )(G + jC ) =  + j = ( j L )( j C ) = j LC = j R = 0, G = 0  =0  =  LC R + j L j L L Then, Z0 = = = Purely real  j L C C Not a function of freq. → Non-dispersive R = 0, G = 0 Also, Constant velocity → Non-dispersive 2 Electromagnetic Measurement & Application (EMMA) Lab  Dispersion An ideal lossless tx-line is non-dispersive (lossy line is always dispersive). 1 Dispersion → Wave(phase) velocity is a function of frequency v p =  (Z 0 ,  ) eye diagram 3 Electromagnetic Measurement & Application (EMMA) Lab  Terminated Lossless Tx-Line Let a lossless tx-line be terminated by a load ZL at z=0 Terminated Lossless lines = I (0 ) Vo + e − j  z V (0 ) = Vo − e + j  z z= V (0 ) = I (0 ) Z L V (0 ) Vo+ + Vo− ZL = = I (0 ) Vo+ Vo− − Z0 Z0 Vo− Z L − Z o Define + =  Vo Z L + Z o Reflection coefficient 4 Electromagnetic Measurement & Application (EMMA) Lab  Terminated Lossless Tx-line How much wave reflected by the load due to mismatch between the load and tx-line ex ) Z L = Z 0   = 0 match Z L  Z 0    0 mismatch Standing-wave is generated due to mismatch vs. Traveling wave Standing wave https://www.walter-fendt.de/html5/phen/standingwavereflection_en.htm 5 Electromagnetic Measurement & Application (EMMA) Lab  Voltage Standing Wave Ratio A mismatch generates standing waves (How much?) V ( z ) = Vo+ e − jz + Vo+ e + jz V ( z ) = Vo+ 1 + e j 2 z ( = Vo+ e − jz 1 + e + j 2 z ) Let  =  e j V ( z ) = Vo+ 1 +  e j ( 2 z + ) + max V ( z ) max = Vo 1+  Polar chart V ( z ) min = Vo+ 1 −  min  1+  V ( z ) max 1+  1−  vswr = swr = = = 1+  V ( z ) min 1−  1−  Matched vswr →1  →0 Total reflection vswr →   →1 6 Electromagnetic Measurement & Application (EMMA) Lab  Gamma, VSWR, RL Z L − Z0 = How much wave is reflected by the load due to the Z L + Z0 impedance mismatch ex) Z L = Z 0   = 0 Perfect match Z L  Z 0    0 Mismatch max V 1+  1+  vswr = max = = V min 1−  1−  min ex) vswr →1  → 0 Perfect match vswr →   →1 Total reflection Return loss negative power RL = −10 log10 (  2 )  dB  = − 20 log 10   RL = + dB Perfect match RL = 0 dB Total reflection 7 Electromagnetic Measurement & Application (EMMA) Lab  Linear Scale vs. Log Scale 𝑿 𝒍𝒐𝒈𝟏𝟎 𝑿 8 https://www.worldometers.info/coronavirus/ Electromagnetic Measurement & Application (EMMA) Lab  deciBel (dB) A log scale used with dimensionless ratios of power over a large dynamic range. Power ( dB ) = 10 log10 Power ( Watt ) Power ( dB ) = 20 log10 Magnitude ( V, A, V m , A m ) 30 dB is an increase of 1000X in power Relative dB 20 dB is an increase of 100X in power dBW, dBm 10 dB is an increase of 10X in power 6 dB is an increase of 4X in power dBV, dBA, dBV/m, dBA/m, 3 dB is an increase of 2X in power dBi, dBd, dBic, dBsm 2 dB is an increase of 1.6X in power 1 dB is an increase of 1.25X in power etc… 0 dB is no increase or decrease in power -1 dB is a decrease of 20% in power 𝟏 𝟏 -2 dB is a decrease of 37% in power (~ 1/3) = 𝟏𝟎 log 𝟏𝟎 𝑷 𝒅𝑩 𝑷 -3 dB is a decrease of 50% in power (~ 1/2) -6 dB is a decrease of 75% in power (~ 3/4) -10 dB is a decrease of 90% in power = −𝟏𝟎 log 𝟏𝟎 𝑷 = −𝑷𝒅𝑩 -20 dB is a decrease of 99% in power -30 dB is a decrease of 99.9% in power 9 Electromagnetic Measurement & Application (EMMA) Lab  Numbers Return loss (dB) VSWR (x : 1) Gamma = - Gamma (dB) 1.00 ∞ 0 1.01 46.06 0.005 1.02 40.09 0.01 1.03 36.61 0.015 1.04 34.15 0.02 1.05 32.26 0.024 1.06 30.71 0.029 1.07 29.42 0.034 1.08 28.3 0.038 1.09 27.32 0.043 1.1 26.44 0.048 1.2 20.83 0.091 1.3 17.69 0.13 1.4 15.56 0.167 1.5 13.98 0.2 1.6 12.74 0.231 1.7 11.73 0.259 1.8 10.88 0.286 1.9 10.16 0.3 2 9.54 0.333 2.5 7.36 0.429 3 6.02 0.5 3.5 5.11 0.556 4 4.44 0.6 4.5 3.93 0.636 5 3.52 0.667 5.5 3.19 0.692 ∞ 0 1 10 Electromagnetic Measurement & Application (EMMA) Lab  Terminated Lossless Tx-Line (Recap.) Let a lossless tx-line is terminated by a load ZL Terminated Lossless lines = I (0 )  → ,   → 0 + V (0 ) = Vo e − jz R = G = 0,  = 0, L  =  LC , Z 0 = , C 1 vp = LC z= How much wave is reflected by the load due to mismatch between load and tx-line ex ) Z L = Z 0   = 0 match Z L  Z 0    0 mismatch 1 Electromagnetic Measurement & Application (EMMA) Lab  Input Impedance (Zin) = I (0 ) V (0 ) = z=-l z= Z in Impedance looking back at z = - l 2 Electromagnetic Measurement & Application (EMMA) Lab  Input Impedance Zin varies with its location.  ex.1) At l=+ 2       2    L +   +   2    L 2   Z jZ tan Z jZ tan   o  0   Z in  z = −  = Z 0  = Z0  = ZL  2  Z + jZ tan      Z + jZ tan  2     0     2    0   2   L L    Also, Zin ( z = − ) = Z L → Zin= ZL in every half-wavelength   2     L Z + jZ tan    Z2    o   4 = 0 ex.2) At l=+ Z in  z = −  = Z 0  4  4  Z + jZ tan  2    Z L  0     4 L  → Quarter wavelength transformer (will learn later) 3 Electromagnetic Measurement & Application (EMMA) Lab  Matched, Short, Open Z L = Z0 ZL = 0 ZL =  4 Electromagnetic Measurement & Application (EMMA) Lab  Short Circuit Terminated by a short circuit → Z L − Z0 standing wave  ( 0) = = −1 Z L + Z0 Total reflection 1+  vswr = → 1−  → Z in ( z = −l ) = jZ o tan (  l ) 5 Electromagnetic Measurement & Application (EMMA) Lab  Open Circuit In case of terminated in an open circuit → Z L − Z0 standing wave  ( 0) = = +1 Z L + Z0 Total reflection 1+  vswr = → 1−  → Z in ( z = −l ) = − jZ o cot (  l ) 6 Electromagnetic Measurement & Application (EMMA) Lab  Short & Open Circuit Short ckt. termination Open ckt. termination voltage current Z in ( z = −l ) = jZ o tan (  l ) Z in ( z = −l ) = − jZ o cot (  l ) impedance 7 Electromagnetic Measurement & Application (EMMA) Lab  Matched, Short, Open Matched open short http://en.wikipedia.org/wiki/Transmission_line 8 Electromagnetic Measurement & Application (EMMA) Lab  Gamma, VSWR, RL (Recap.) Return loss (dB) VSWR (x : 1) Gamma = - Gamma (dB) 1.00 ∞ 0 1.01 46.06 0.005 1.02 40.09 0.01 1.03 36.61 0.015 1.04 34.15 0.02 1.05 32.26 0.024 1.06 30.71 0.029 Z L − Z0 1.07 29.42 0.034 = Z L + Z0 1.08 28.3 0.038 1.09 27.32 0.043 1+  1.1 26.44 0.048 vswr = 1.2 20.83 0.091 1−  1.3 17.69 0.13 1.4 15.56 0.167 RL= − 20 log10  1.5 13.98 0.2 1.6 12.74 0.231 1.7 11.73 0.259 1.8 10.88 0.286 Lossy tx-line (dispersive tx-line) 1.9 10.16 0.3    ,    0 2 9.54 0.333 2.5 7.36 0.429 R  0, G  0 3 6.02 0.5   0, Z 0 &  are complex numbers 3.5 5.11 0.556 4 4.44 0.6 4.5 3.93 0.636 5 3.52 0.667 5.5 3.19 0.692 ∞ 0 1 1 Electromagnetic Measurement & Application (EMMA) Lab  Example of Coaxial Cable Loss https://www.maximintegrated.com/en/design/technical-documents/app-notes/4/4303.html 2 Electromagnetic Measurement & Application (EMMA) Lab  Note A good tx-line for energy transfer should have low attenuation, low vswr (or low  or high return loss). Use conductor with high conductivity Use dielectric with low dielectric loss Match the load impedance to tx-line Z0 Let’s learn how to analyze impedance matching conditions and how to improve it USING SMITH CHART 3 Electromagnetic Measurement & Application (EMMA) Lab  Smith Chart 4 Electromagnetic Measurement & Application (EMMA) Lab  Impedance (Recap.) V Z= = R + jX = R + j ( X L − X C ) I  1  = R + j  L −    C  Wave impedance (Intrinsic impedance) E   Input impedance ZW =  = = = H k  ZS Characteristic impedance Source Vo+ R + j L ZL Z0 = + = I o G + j C 5 Electromagnetic Measurement & Application (EMMA) Lab  Smith Chart 1939 by P. W. Smith A graphical tool to analyze and design tx-line circuits Smith chart is simply a 2D complex plane in polar form. It is useful to locate complex parameters such as  =  e j = r + j i Z = Z e j  z = R + jX X Z = R 2 + X 2 ;  Z = tan −1 R Z 0 = R0 + jX 0 Z L − Z0 Z L = RL + j X L = Z L + Z0 Zin = Rin + j X in 6 Electromagnetic Measurement & Application (EMMA) Lab  Smith Chart Smith Chart maps rectilinear impedance plane onto polar plane (Infinite plane) (Finite diagram) Keysight, Network Analyzer Basics 7 Electromagnetic Measurement & Application (EMMA) Lab  Smith Chart http://en.wikipedia.org/wiki/Smith_chart Z in -plane z-plane (normalized by Z0) ZL z −1 1+  z= =r+ jx = z=  =  e j = r + j i Z0 z +1 1−  8 Electromagnetic Measurement & Application (EMMA) Lab  rL and xL Circles Z L Z0 − ZL Z Z 0 zL − 1 Normalize ZL by Z0 zL = , = 0 = Z0 Z L Z 0 zL + 1 + Z0 Z0 1+  1 + r + ji Plug in zL = rL + jx L = 1−  1 − r − ji z L = rL + j xL  = r + j i Multiplying complex conjugate rL circle re-write xL circle 9 Electromagnetic Measurement & Application (EMMA) Lab  rL Circles 2 2  1 1 rL = 1,  r −  + i =   2  2 2 2 2  3 1 rL = 3,   r −  +  i 2 =    4 4 2 2  3   10  rL = 0.3,   r −  +  i 2 =    13   13  10 Electromagnetic Measurement & Application (EMMA) Lab  xL Circles x L = 1, (r − 1)2 + (i − 1)2 = 1 xL =  1 , (r − 1)2 + (i  2)2 = 2 2 2 2 2  1 1 xL = 2 , (r − 1) +  i   =   2  2 2 11 Electromagnetic Measurement & Application (EMMA) Lab  Combination 12 Electromagnetic Measurement & Application (EMMA) Lab  Regions by r Invalid (no negative r) smaller than Z0 Invalid (no negative r) larger than Z0 ZL z= =r+ jx Z0 Match to Z0  =  e j = r + j i 13 Electromagnetic Measurement & Application (EMMA) Lab  Regions by x Z = R + jX = R + j ( X L − XC ) inductive X L = j L  1  = R + j  L −    C  ZL z= =r+ jx Z0  =  e j = r + j i capacitive 1 XC = j C 14 Electromagnetic Measurement & Application (EMMA) Lab  Combination 15 Electromagnetic Measurement & Application (EMMA) Lab  Smith Chart Z z= =r+ jx  =  e j = r + j i Z0 1 Electromagnetic Measurement & Application (EMMA) Lab  Phase Shift z = -l z= Z in  Z L + jZ 0 tan  l   1 + e − j 2  l  Recall Z in ( z = −l ) = Z 0   = Z 0  − j 2 l   0 Z + jZ L tan  l   1 −  e  j − j 2  l Z in ( z = −l ) 1 +  e e zin ( z = −l ) = = Z0 1 −  e j e − j 2  l j 1+  1+  e Difference is 2 l compare with z L = = 1 −  1 −  e j phase shift in  2 Electromagnetic Measurement & Application (EMMA) Lab  Phase Shift (z = zin ) Phase shift along the Gamma circle 2 l ( z = z L ) 3 Electromagnetic Measurement & Application (EMMA) Lab  Phase Shift One turn: Half-wavelength Half turn: Quarter-wavelength  2  l= → l== 2  2  2   l= → l= = 4  4 2 z =1 =0 Z L = Z0 short z=0 open z=  = −1  =1 4 Electromagnetic Measurement & Application (EMMA) Lab  Short & Open Circuit (Recap.) Short ckt. termination Open ckt. termination voltage current Z in ( z = −l ) = jZ o tan (  l ) Z in ( z = −l ) = − jZ o cot (  l ) impedance 5 Electromagnetic Measurement & Application (EMMA) Lab  Frequency Response  1  Z ( ) = R ( ) + jX ( ) = R ( ) + j   L −    C  1 GHz 5 GHz Rotates along the clockwise direction from low to high frequency 6 Electromagnetic Measurement & Application (EMMA) Lab  VSWR Circles 7 Electromagnetic Measurement & Application (EMMA) Lab  Impedance & Admittance Charts V Z= = R + jX I  1  = R + j  L −     C  I 1 Y= = = G + jB  S  V Z Admittance Conductance Susceptance Why using admittance? → useful to express shunt (or parallel) circuit 8 Electromagnetic Measurement & Application (EMMA) Lab  Impedance & Admittance Chart 9 Electromagnetic Measurement & Application (EMMA) Lab  Smith Chart Exercise 百聞不如一見 A thousand hearings are not worth one seeing. Free Smith chart tools: http://www.fritz.dellsperger.net/smith.html https://www.will-kelsey.com/smith_chart/ https://smithchart.net/ http://www.ae6ty.com/Smith_Charts.html 1 Electromagnetic Measurement & Application (EMMA) Lab  Example 2 Electromagnetic Measurement & Application (EMMA) Lab  Example 3 Electromagnetic Measurement & Application (EMMA) Lab  Example 4 Electromagnetic Measurement & Application (EMMA) Lab  Example Here is the 50 ohm normalized Smith chart 1) Draw g = 1/50 ohm circle 2) Draw 25 ohm curve 3) Mark the open 4) The locus shows the reflection coefficient as the frequency changes. Mark the starting and ending frequencies. 5 Electromagnetic Measurement & Application (EMMA) Lab  Matching Example Match a 100 W antenna to a 50 W transmission line at 100 MHz 6 Electromagnetic Measurement & Application (EMMA) Lab  Locus of L and C Z = j L Z0 j L Z0 1 1 j C Z0 Z0 j C 7 Electromagnetic Measurement & Application (EMMA) Lab  Antenna Impedance Example +j1.0 +j0.5 +j2.0 Stub length +j0.2 +j5.0 0.2 0.5 1.0 2.0 5.0 0 0.0  -5 (dB) (dB) -10 -j0.2 -j5.0 11 SG xfy=0.7 mm -15 xfy=0.8 mm -20 -j0.5 -j2.0 -25 60 65 70 75 80 85 90 95 100 Frequency (GHz) -j1.0 8 Electromagnetic Measurement & Application (EMMA) Lab  Why Z0 = 50 Ohm? For a coaxial cable, min attenuation at 77 ohm max power transfer at 30 ohm. → arithmetic mean 53.5 ohm geometrical mean 48 ohm → overall, 50 ohm Standard needed → 50 ohm Trade-off between Power handling capacity & Attenuation 9 Electromagnetic Measurement & Application (EMMA) Lab  Note (Recap.) A good tx-line for energy transfer should have low attenuation, low vswr (or low G or high return loss). Use conductor with high conductivity Use dielectric with low dielectric loss Match the load impedance to tx-line Z0 Let’s learn how to analyze impedance matching conditions and how to improve it USING SMITH CHART 1 Electromagnetic Measurement & Application (EMMA) Lab  Locus of L and C in Smith Chart (Recap.) Z = j L Z0 j L Z0 1 1 j C Z0 Z0 j C 2 Electromagnetic Measurement & Application (EMMA) Lab  Impedance Matching Network Z = R + jX = R + j ( X L − X C )  1  = R + j  L −    C  R + j L j L L Z0 =  =  j L C C Lossless tx-line 3 Electromagnetic Measurement & Application (EMMA) Lab  Matching Network Matching network (or circuit) design considerations: → Operation frequency (high? low? narrow? broad? multi?) → Distance between the load and source (tx-line length) → Area of implementation, simplicity, cost 4 Electromagnetic Measurement & Application (EMMA) Lab  Conjugate & Reflectionless Matching ZS Z S = RS + jX S P W Z L = RL + jX L G ZL ZL − ZS V G= ZL + ZS Conjugate matching Reflectionless matching - Power matching (Power waves) - Impedance matching (Voltage waves) - Maximum power transfer - No reflection XL = −XS X = XS ZL = Z *  ,G0 ZL = ZS  L , G=0 S  L R = RS  RL = RS 2 2 V V Pmax = P = Re ( Z S ) 4 Re ( Z S ) 2 4 ZS Pmax  P , if Im ( Z L )  0 5 Electromagnetic Measurement & Application (EMMA) Lab  Conjugate & Reflectionless Matching Conjugate matching Reflectionless matching ZO 6 Electromagnetic Measurement & Application (EMMA) Lab  Conjugate & Reflectionless Matching Which matching should be concerned? → For a narrow band application w/ short tx-line: Conjugate matching → For a broadband application w/ long tx-line: Reflectionless matching For microwave engineering, reflectionless matching is preferred since → typically an electrically long low-loss tx-line is used. → impedances vary with a slight change of time, location, and frequency. → most microwave applications require a certain amount of bandwidth. Assuming a lossless tx-line → Z0 is real → Reflectionless matching = Conjugate matching 7 Electromagnetic Measurement & Application (EMMA) Lab  Lumped vs. Distributed Element Matching Pros: Easy to switch (soldering) Pros: Good reliability, repeatability Cons: Discrete values Cons: Too big at low frequency High loss at high freq. Not easy to tune Self resonant freq. (SRF) Distributed element matching Lumped element matching (Tx-line element matching) Frequency few kHz ~ few GHz few GHz ~ Reliability Bad Good Tunability Good Bad www.rfdh.com 8 Electromagnetic Measurement & Application (EMMA) Lab  Lumped Elements (RLC) 9 Electromagnetic Measurement & App