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## Chapter 2. Transmission Line Theory

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**Chapter 2. Transmission Line Theory**Sept. 29th, 2008**2.1 Transmission Lines**• A transmission line is a distributed-parameter network, where voltages and currents can vary in magnitude and phase over the length of the line. Lumped Element Model for a Transmission Line • Transmission lines usually consist of 2 parallel conductors. • A short segment Δz of transmission line can be modeled as a lumped-element circuit.**Figure 2.1 Voltage and current definitions and equivalent**circuit for an incremental length of transmission line. (a) Voltage and current definitions. (b) Lumped-element equivalent circuit.**R = series resistance per unit length for both conductors**• L = series inductance per unit length for both conductors • G = shunt conductance per unit length • C = shunt capacitance per unit length • Applying KVL and KCL,**Dividing (2.1) by Δz and Δz 0,** Time-domain form of the transmission line, or telegrapher, equation. • For the sinusoidal steady-state condition with cosine-based phasors,**Wave Propagation on a Transmission Line**• By eliminating either I(z) or V(z): where the complex propagation constant. (α = attenuation constant, β = phase constant) • Traveling wave solutions to (2.4): Wave propagation in +z directon Wave propagation in -z directon**Applying (2.3a) to the voltage of (2.6),**• If a characteristic impedance, Z0, is defined as • (2.6) can be rewritten**Converting the phasor voltage of (2.6) to the time domain:**• The wavelength of the traveling waves: • The phase velocity of the wave is defined as the speed at which a constant phase point travels down the line,**Lossless Transmission Lines**• R = G = 0 gives or • The general solutions for voltage and current on a lossless transmission line:**The wavelength on the line:**• The phase velocity on the line:**2.2 Field Analysis of Transmission Lines**• Transmission Line Parameters Figure 2.2 (p. 53)Field lines on an arbitrary TEM transmission line.**The time-average stored magnetic energy for 1 m section of**line: • The circuit theory gives • Similarly, **Power loss per unit length due to the finite conductivity**(from (1.130)) • Circuit theory (H || S) • Time-average power dissipated per unit length in a lossy dielectric (from (1.92))**Circuit theory **• Ex 2.1 Transmission line parameters of a coaxial line • Table 2.1**The Telegrapher Equations Derived form Field Analysis of a**Coaxial Line • Eq. (2.3) can also be obtained from ME. • A TEM wave on the coaxial line: Ez = Hz = 0. • Due to the azimuthal symmetry, no φ-variation ə/əφ = 0 • The fields inside the coaxial line will satisfy ME. where**Since the z-components must vanish,**From the B.C., Eφ = 0 at ρ = a, b Eφ = 0 everywhere**The voltage between 2 conductors**The total current on the inner conductor at ρ = a**Propagation Constant, Impedance, and Power Flow for the**Lossless Coaxial Line • From Eq. (2.24) • For lossless media, • The wave impedance • The characteristic impedance of the coaxial line Ex 2.1**Power flow ( in the z direction) on the coaxial line may be**computed from the Poynting vector as • The flow of power in a transmission line takes place entirely via the E & H fields between the 2 conductors; power is not transmitted through the conductors themselves.**2.3 The Terminated Lossless Transmission Lines**The total voltage and current on the line**The total voltage and current at the load are related by the**load impedance, so at z = 0 • The voltage reflection coefficient: • The total voltage and current on the line:**It is seen that the voltage and current on the line consist**of a superposition of an incident and reflected wave. standing waves • WhenΓ= 0 matched. • For the time-average power flow along the line at the point z:**When the load is mismatched, not all of the available power**from the generator is delivered to the load. This “loss” is return loss (RL): RL = -20 log|Γ| dB • If the load is matched to the line, Γ= 0 and |V(z)| = |V0+| (constant) “flat”. • When the load is mismatched,**A measure of the mismatch of a line, called the voltage**standing wave ratio (VSWR) (1< VSWR<∞) • From (2.39), the distance between 2 successive voltage maxima (or minima) is l = 2π/2β = λ/2 (2βl = 2π), while the distance between a maximum and a minimum is l = π/2β = λ/4. • From (2.34) with z = -l,**At a distance l = -z,** Transmission line impedance equation**Special Cases of Terminated Transmission Lines**• Short-circuited line ZL = 0 Γ= -1**Figure 2.6 (a) Voltage, (b) current, and (c) impedance (Rin**= 0 or ) variation along a short-circuited transmission line.**Open-circuited line**ZL = ∞ Γ= 1**Figure 2.8 (a) Voltage, (b) current, and (c) impedance (Rin**= 0 or ) variation along an open-circuited transmission line.**Terminated transmission lines with special lengths.**• If l = λ/2, Zin = ZL. • If the line is a quarter-wavelength long, or, l = λ/4+ nλ/2 (n = 1,2,3…), Zin = Z02/ZL. quarter-wave transformer**Figure 2.9 (p. 63)Reflection and transmission at the**junction of two transmission lines with different characteristic impedances.**2.4 The Smith Chart**• A graphical aid that is very useful for solving transmission line problems. Derivation of the Smith Chart • Essentially a polar plot of the Γ(= |Γ|ejθ). • This can be used to convert from Γto normalized impedances (or admittances), and vice versa, using the impedance (or admittance) circles printed on the chart.**If a lossless line of Z0 is terminated with ZL, zL = ZL/Z0**(normalized load impedance), • Let Γ= Γr +jΓi, and zL = rL + jxL.**The Smith chart can also be used to graphically solve the**transmission line impedance equation of (2.44). • If we have plotted |Γ|ejθ at the load, Zin seen looking into a length l of transmission line terminates with zLcan be found by rotating the point clockwise an amount of 2βl around the center of the chart.**Smith chart has scales around its periphery calibrated in**electrical lengths, toward and away from the “generator”. • The scales over a range of 0 to 0.5 λ.**Ex 2.2 ZL = 40+j70,l = 0.3λ, find Γl, Γin and Zin**Figure 2.11 (p. 67)Smith chart for Example 2.2.**The Combined Impedance-Admittance Smith Chart**• Since a complete revolution around the Smith chart corresponds to a line length of λ/2, a λ/4 transformation is equivalent to rotating the chart by 180°. • Imaging a give impedance (or admittance) point across the center of the chart to obtain the corresponding admittance (or impedance) point.**Ex 2.3ZL = 100+j50, YL, Yin ? when l = 0.15λ**Figure 2.12 (p. 69)ZY Smith chart with solution for Example 2.3.**The Slotted Line**• A transmission line allowing the sampling of E field amplitude of a standing wave on a terminated line. • With this device the SWR and the distance of the first voltage minimum from the load can be measured, from this data ZL can be determined. • ZL is complex 2 distinct quantities must be measured. • Replaced by vector network analyzer.**Assume for a certain terminated line, we have measured the**SWR on the line and lmin, the distance from the load to the first voltage minimum on the line. • Minimum occurs when • The phase of Γ = • Load impedance**Ex 2.4**• With a short circuit load, voltage minima at z = 0.2, 2.2, 4.2 cm • With unknown load, voltage minima at z = 0.72, 2.72, 4.72 cm • λ = 4 cm, • If the load is at 4.2 cm, lmin = 4.2 – 2.72 = 1.48 cm = 0.37 λ**Figure 2.14 (p. 71)Voltage standing wave patterns for**Example 2.4. (a) Standing wave for short-circuit load. (b) Standing wave for unknown load.**2.5 The Quarterwave Transformer**Impedance Viewpoint • For βl = (2π/λ)(λ/4) = π/2 • In order for Γ = 0, Zin = Z0**Ex 2.5 Frequency Response of a Quarter-Wave Transformer**• RL = 100, Z0 = 50**Figure 2.17 (p. 74)Reflection coefficient versus normalized**frequency for the quarter-wave transformer of Example 2.5.**The Multiple Reflection Viewpoint**Figure 2.18 (p. 75)Multiple reflection analysis of the quarter-wave transformer.