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

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.

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

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

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!

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

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+

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 vp depends on equivalent circuit parameters related to the structure and with line length (l) determines the time/phase delay

Circuit Model versus Field Model for 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.

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

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]

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

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.

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

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

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

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

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

Parameters of the T-line Wave Functions

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

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!

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!

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

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

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!

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

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

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

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

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)

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