Electromagnetics (ENGR 367)

1. Electromagnetics (ENGR 367) Transmission Lines (T-lines)

2. Introduction to T-lines • Function of T-line: to carry wave energy from one location to another • T-line terminology • origin of waves: source (e.g. generator) • destination of waves: load (e.g. receiving device) • Value of transmitted electrical wave energy • provides light, heat or mechanical work, etc. • carries signal information • Audio: speech or music • Visual images: static or dynamic, real-time or replay • Data: computer, telemetry system, financial activity, etc.

3. Examples of T-lines • Coax connection between the power amplifier and antenna of an RF broadcast system • Fiber optic cable links between networked computers • Power line connection between a generating plant and a distant substation • Connection between a cable TV service provider and a consumer’s set • Trace connections between devices on a PCB operating at HF

4. What can electrical engineers understand and know how to do with Wave Phenomena on T-lines? • Treat them as circuit elements with a complex impedance that depends on length (l) and frequency (=2f) • Model wave propagation on them that behave as lossy, low loss, or approximately lossless • Handle multiple line sections that connect to split power, match impedance, etc. • Account for transient phenomena in T-lines in effect when they carry pulse/digital data

5. Extraordinary Feature of T-lines • While the circuit model of a T-line includes parameters that depend on length, T-lines have a unique characteristic impedance independent of length! How can this be? • We start with two assumptions that take us beyond traditional circuit analysis!

6. Two Assumptions:T-Line Theory vs. Circuit Analysis • If connection distance (d) between devices is • on the order of a wavelength or more (d > ~), then phase differences between devices may be appreciable and wave phenomena becomes significant • d << , then basic circuit analysis methods will suffice • If the dimension (D) of a circuit element from its input to output is • large compared to a wavelength (D >> ) then significant propagation time can exist through it and the element should be treated as distributed (i.e., using R,L,C,G/unit length) • D < , then a lumped (ideal) element approximation is OK

7. Basic T-line Concepts • Many practical T-lines may be modeled approximately as a two-wire line • Closing the switch launches a wave-front from source (e.g., battery) to load (e.g. resistor, R) • The wave-front may be characterized by • Voltage V+ = V0 • Current I+

8. Basic T-line Concepts • Practical T-line Modeled as a Two-wire line • V+, I+ wave-fronts travel at finite wave velocity (vp<c) so that voltages and currents along the line do not change instantaneously • vpdepends on equivalent circuit parameters related to the structure and with line length (l)determines the time/phase delay

9. Circuit Model versus Field Modelfor Wave Propagation on T-lines • Circuit model: identifies equivalent circuit parameters for T-line and treats it in terms of voltage (V) and current (I) • Field model: applies Maxwell’s equations to line configuration to get functions for E, H followed by expressions for power (P), wave velocity (vp), etc.

10. T-line Circuit versus Field Model:Applicability • Field model: a better approximation at high frequency (HF) and more useful to predict loss, complicated wave behavior • Circuit model: a better approximation at low frequency (LF) and simpler, so we will focus on this model for now

11. T-line Theory: Circuit Model • Static electric and magnetic field analysis shows that • each real conducting wire by itself has • per unit length resistance R [/m] (ohmic loss) • per unit length inductance L [H/m] • two conducting wires separated from each other by a practical dielectric insulator have • per unit length conductance G [S/m] (leakage loss) • per unit length capacitance C [F/m]

12. Equivalent Lumped Element Circuit Model Short T-line Section z

13. Equivalent Impedance-Admittance Circuit Model Infinitessimal T-line Section dz where Zs = R+jL[/m] and Yp = G+jC [S/m] under the condition of time-harmonic osc.

14. Derive the T-line Wave Equations • Treat voltage and current as time dependent phasor functions where • By Ohm’s Law applied to the T-line section dz

15. Derive the T-line Wave Equations • Differentiating 1) and 2) w/r/to z and putting both terms on the LHS

16. Derive the T-line Wave Equations • Substituting 2) into 3): • Substituting 1) into 4): Two simultaneous 2nd order differential equations

17. Solutions to T-line Wave Equations in Complex Exponential Form • For the voltage function: • For the current function:

18. Recall Euler’s Identity • Vital to understanding the wave functions • Shows how to find cos & sin functions in terms of their complex exponential counterparts

19. Parameters of the T-line Wave Functions

20. Explicit T-line Wave Functions in terms of  and  • The Voltage function: • The Current function: since  =  + j 

21. Other Essential T-line Parameters • Characteristic Impedance (Z0)≡ratio of voltage to current anywhere along the line • from the circuit model with loss components, we have • thus the general characteristic impedance is Note: Z0 is independent of length!

22. Other Essential T-line Parameters • Wave Propagation (Phase) Velocity (vp) • in terms of the basic wave parameters • and from the circuit model including loss components Note: expressions for Z0 and vp simplify in lossless case!

23. Lossless T-line • Assumptions: R = 0, G = 0 • the characteristic impedance becomes • the propagation velocity becomes

24. Low-loss T-line Approximation • Assumptions: R << L, G << C • Using the first three terms of the binomial series

25. Low-loss T-line Approximation • Attenuation: • Phase Constant: • Characteristic Impedance: • Propagation Velocity: Note: expressions for , , Z0, and vp in terms of , R, L, G and C left for you to work out as HW!

26. Example of Calculating T-line Wave Parameters from Circuit Parameters • Exercise (D11.1 from Hayt & Buck, 7/e, p. 347.) Given: an operating frequency of 500 Mrad/s and T-line circuit values of R = 0.2 /m, L = 0.25 H/m, G = 10 S/m, and C = 100 pF/m. Find: values for , , , vp and Z0 Solution: 1st check for the validity of any approximation

27. Example of Calculating T-line Wave Parameters from Circuit Parameters • Exercise (D11.1 continued) Solution: lossless approximation good for everything except  so

28. Summary • T-lines carry wave energy over distances valuable in RF broadcast, computer, cable TV, power and other HF applications • If the transmission distance and element dimensions are significant compared to a wavelength, then T-lines exhibit wave phenomena and distributed element behavior

29. Summary • Many practical T-lines act like a two-wire line with voltage and current wave-fronts that propagate at finite speed • The circuit model of a T-line, applicable at lower frequencies, includes per unit length resistance (R), inductance (L), capacitance (C) and conductance (G) that lead to wave equations for voltage and current

30. Summary • T-line wave equations are satisfied by complex exponential functions for voltage and current representing forward and backward sinusoidal traveling waves • Lossy, low-loss or lossless T-lines may be described by parameters including phase constant (), attenuation (), wavelength (), propagation velocity (vp) and characteristic impedance(Z0)

31. References • Hayt & Buck, Engineering Electromagnetics, 7/e, McGraw Hill: New York, 2006. • Kraus & Fleisch, Electromagnetics with Applications, 5/e, McGraw Hill: New York, 1999.