ECE 546 – Jose Schutt-Aine 1 ECE 546 Lecture -04 Transmission Lines Spring 2014 Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois

ECE 546 – Jose Schutt-Aine 2 Faraday’s Law of Induction Ampère’s Law Gauss’ Law for electric field Gauss’ Law for magnetic field Constitutive Relations Maxwell’s Equations

ECE 546 – Jose Schutt-Aine 3 z Wavelength : = propagation velocity frequency Why Transmission Line?

ECE 546 – Jose Schutt-Aine 4 In Free Space At 10 KHz : = 30 km At 10 GHz : = 3 cm Transmission line behavior is prevalent when the structural dimensions of the circuits are comparable to the wavelength. Why Transmission Line?

ECE 546 – Jose Schutt-Aine 5 Let d be the largest dimension of a circuit If d << , a lumped model for the circuit can be used circuit z Justification for Transmission Line

ECE 546 – Jose Schutt-Aine 6 circuit z If d ≈, or d > then use transmission line model Justification for Transmission Line

ECE 546 – Jose Schutt-Aine 7 Short Lumped Reactive CKT Transmission Line Low Frequency Mid-range Frequency High Frequency or Modeling Interconnections

ECE 546 – Jose Schutt-Aine 8 Single wire near ground

ECE 546 – Jose Schutt-Aine 9 Single wire between grounded parallel planes ground return

ECE 546 – Jose Schutt-Aine 10 Wires in parallel near ground for d << D, h

ECE 546 – Jose Schutt-Aine 11 Balanced, near ground for d << D, h

ECE 546 – Jose Schutt-Aine 12 Open 2-wire line in air

ECE 546 – Jose Schutt-Aine 13 Parallel-plate Transmission Line

ECE 546 – Jose Schutt-Aine 14 Types of Transmission Lines

ECE 546 – Jose Schutt-Aine 15 Coaxial Transmission Line TEM Mode of Propagation

ECE 546 – Jose Schutt-Aine 16 Coaxial Air Lines Infinite Conductivity Finite Conductivity

ECE 546 – Jose Schutt-Aine 17 Coaxial Connector Standards ConnectorFrequency Range 14 mmDC GHz GPC-7DC - 18 GHz Type NDC - 18 GHz 3.5 mmDC - 33 GHz 2.92 mmDC - 40 GHz 2.4 mmDC - 50 GHz 1.85 mmDC - 65 GHz 1.0 mmDC GHz

ECE 546 – Jose Schutt-Aine 18 Microstrip

ECE 546 – Jose Schutt-Aine 19 Microstrip h = 21 mils h = 14 mils h = 7 mils Microstrip Characteristic Impedance W/h Zo (ohms) dielectric constant : 4.3.

ECE 546 – Jose Schutt-Aine 20 L: Inductance per unit length. C: Capacitance per unit length. Assume time-harmonic dependence Telegraphers’ Equations

ECE 546 – Jose Schutt-Aine 21 TL Solutions

ECE 546 – Jose Schutt-Aine 22 forward wave backward wave (Frequency Domain) TL Solutions

ECE 546 – Jose Schutt-Aine 23 Propagation constant Characteristic impedance Wavelength Propagation velocity TL Solutions

ECE 546 – Jose Schutt-Aine 24 At z=0, we have V(0)=Z R I(0) But from the TL equations: Which gives is the load reflection coefficientwhere Reflection Coefficient

ECE 546 – Jose Schutt-Aine 25 - If Z R = Z o,  R =0, no reflection, the line is matched - If Z R = 0, short circuit at the load,  R =-1 - If Z R  inf, open circuit at the load,  R =+1 Reflection Coefficient V and I can be written in terms of  R

ECE 546 – Jose Schutt-Aine 26 Generalized Impedance Impedance transformation equation

ECE 546 – Jose Schutt-Aine 27 - Short circuit Z R =0, line appears inductive for 0 < l < /2 - Open circuit Z R  inf, line appears capacitive for 0 < l < /2 - If l = /4, the line is a quarter-wave transformer Generalized Impedance

ECE 546 – Jose Schutt-Aine 28 Reflection coefficient transformation equation Generalized Reflection Coefficient

ECE 546 – Jose Schutt-Aine 29 Maximum and minimum magnitudes given by Voltage Standing Wave Ratio (VSWR) We follow the magnitude of the voltage along the TL

ECE 546 – Jose Schutt-Aine 30 Voltage Standing Wave Ratio (VSWR) Define Voltage Standing Wave Ratio as: It is a measure of the interaction between forward and backward waves

ECE 546 – Jose Schutt-Aine 31 VSWR – Arbitrary Load Shows variation of amplitude along line

ECE 546 – Jose Schutt-Aine 32 VSWR – For Short Circuit Load Voltage minimum is reached at load

ECE 546 – Jose Schutt-Aine 33 Voltage maximum is reached at load VSWR – For Open Circuit Load

ECE 546 – Jose Schutt-Aine 34 VSWR – For Open Matched Load No variation in amplitude along line

ECE 546 – Jose Schutt-Aine 35 Application: Slotted-Line Measurement -Measure VSWR = V max /V min -Measure location of first minimum

ECE 546 – Jose Schutt-Aine 36 Application: Slotted-Line Measurement At minimum, Therefore, So, Since then and

ECE 546 – Jose Schutt-Aine 37 Summary of TL Equations Impedance Transformation  Reflection Coefficient Transformation  Reflection Coefficient – to Impedance  Impedance to Reflection Coefficient  Voltage Current

ECE 546 – Jose Schutt-Aine 38 At z = -l, Determining V + For lossless TL, V and I are given by reflection coefficient at the load

ECE 546 – Jose Schutt-Aine 39 this leads to or Determining V +

ECE 546 – Jose Schutt-Aine 40 Divide through by withand From which Determining V +

ECE 546 – Jose Schutt-Aine 41 A signal generator having an internal resistance Z s = 25  and an open circuit phasor voltage V s = 1e j0 volt is connected to a 50-  lossless transmission line as shown in the above picture. The load impedance is Z R = 75  and the line length is /4. Find the magnitude and phase of the load current I R. TL Example

ECE 546 – Jose Schutt-Aine 42 TL Example – Cont’

ECE 546 – Jose Schutt-Aine 43 TL Example – Cont’

ECE 546 – Jose Schutt-Aine 44 Geometric Series Expansion V + can be expanded in a geometric series form Since

ECE 546 – Jose Schutt-Aine 45 at z=0 TL Time-Domain Solution

ECE 546 – Jose Schutt-Aine 46 At z=-l TL Time-Domain Solution

ECE 546 – Jose Schutt-Aine 47 TL - Time-Domain Reflectometer For TDR, Z S = Z o   S = 0, and retain only k=1

ECE 546 – Jose Schutt-Aine 48 Forward traveling wave at port 1 (measured at near end of line) Forward traveling wave at port 2 (measured at far end of line) Backward traveling wave at port 1 (measured at near end of line) Backward traveling wave at port 2 (measured at far end of line) Wave Shifting Method

ECE 546 – Jose Schutt-Aine 49 Wave Shifting Solution* *Schutt-Aine & Mittra, Trans. Microwave Theory Tech., pp , vol. 36 March 1988.

ECE 546 – Jose Schutt-Aine 50 Frequency Dependence of Lumped Circuit Models –At higher frequencies, a lumped circuit model is no longer accurate for interconnects and one must use a distributed model –Transition frequency depends on the dimensions and relative magnitude of the interconnect parameters.

ECE 546 – Jose Schutt-Aine 51 Lumped Circuit or Transmission Line? Determine frequency or bandwidth of signal –RF/Microwave: f= operating frequency –Digital: f=0.35/t r Determine the propagation velocity and wavelength –Material medium v=c/(  r ) 1/2 –Obtain wavelength =v/f Compare wavelenth with feature size –If >> d, use lumped circuit: L tot = L* length, C tot = C* length –If  10d or <10d, use transmission-line model

ECE 546 – Jose Schutt-Aine 52 PCB line10 in> 55 MHz< 7ns Package1 in> 400 MHz< 0.9 ns VLSI int*100 um> 8 GHz< 50 ps LevelDimensionFrequencyEdge rate * Using RC criterion for distributed effect Frequency Dependence of Lumped Circuit Models