Presentation on theme: "EKT241 – ELECTROMAGNETICS THEORY"— Presentation transcript:
1 EKT241 – ELECTROMAGNETICS THEORY Chapter 5 Transmission Lines
2 Chapter Objectives Introduction to transmission lines Lump-element model that represent TEM linesLossless lineSmith Chart to analyze transmission line problem
3 Chapter Outline 5-1) General Considerations Lumped-Element Model Transmission-Line EquationsWave Propagation on a Transmission LineThe Lossless Transmission LineInput Impedance of the Lossless LineSpecial Cases of the Lossless LinePower Flow on a Lossless Transmission LineThe Smith ChartImpedance MatchingTransients on Transmission Lines5-2)5-3)5-4)5-5)5-6)5-7)5-8)5-9)5-10)5-11)
4 5-1 General Considerations Transmission lines connect a generator circuit to a load circuit at the receiving end.Transverse electromagnetic (TEM) lines have waves that propagate transversely.
5 5-2 Lumped-Element Model Transmission lines can be represented by a lumped-element circuit model.
6 5-2 Lumped-Element Model Lumped-element circuit model consists 4 transmission line parameters:R’ (Ω/m)L’ (H/m)G’ (S/m)C’ (F/m)
7 5-2 Lumped-Element Model In summary,All TEM transmission lines share the relations:where µ, σ, ε = properties of conductor
8 5-3 Transmission-Line Equations Transmission line equations in phasor form is given as
9 5-4 Wave Propagation on a Transmission Line The wave equation is derived asγ has real part α (attenuation constant) and imaginary part β (phase constant).Complex propagation constant
10 5-4 Wave Propagation on a Transmission Line Characteristic impedance Z0 of the line isPhase velocity for propagating wave iswhere f = frequency (Hz)λ = wavelength (m)
11 Example 5.1 Air LineAn air line is a transmission line for which air is thedielectric material present between the two conductors, which renders G’ = 0. In addition, the conductors are made of a material with high conductivity so that R’ ≈0. For an air line with characteristic impedance of 50 and phase constant of 20 rad/m at 700 MHz, find the inductance per meter and the capacitance per meter of the line.
12 Solution 5.1 Air Line The following quantities are given: With R’ = G’ = 0,The ratio is given byWe get L’ from Z0
13 5-5 The Lossless Transmission Line Low R’ and G’ for transmission line is called lossless transmission line.Using relation properties,
14 5-5 The Lossless Transmission Line Wavelength is given bywhere εr = relative permittivityFor the lossless line, there are 2 unknowns in the equations for the total voltage and current on the line.
15 5-5.1 Voltage Reflection Coefficient The relations for lossless areA load that is matched to the line when ZL = Z0, Γ = 0 and V0−= 0.
16 Example 5.2 Reflection Coefficient of a Series RC Load A 100-Ω transmission line is connected to a loadconsisting of a 50-Ω resistor in series with a 10-pFcapacitor. Find the reflection coefficient at the load fora 100-MHz signal.
17 Solution 5.2 Reflection Coefficient of a Series RC Load The following quantities are givenThe load impedance isVoltage reflection coefficient is
18 5-5.2 Standing Waves 3 types of voltage standing-wave patterns: (a) Matched load(b) Short-circuited line(c) Open-circuited line
19 Standing WavesTo find maximum and minimum values of voltage magnitude, we have
20 5-5.2 Standing Waves First voltage maximum occurs at First voltage minimum occurs atVoltage standing-wave ratio S is defined as
21 Example 5.4 Standing-wave Ratio A 50- transmission line is terminated in a load with ZL = (100 + j50)Ω . Find the voltage reflection coefficient and the voltage standing-wave ratio (SWR).We have,S is given bySolution
22 5-6 Input Impedance of the Lossless Line Voltage to current ratio is called input impedance Zin.The input impedance at z = −l is given asand
23 Example 5.6 Complete Solution for v(z, t) and i(z, t) A 1.05-GHz generator circuit with series impedanceZg = 10Ω and voltage source given byis connected to a load ZL = (100 + j50) through a50-Ω, 67-cm-long lossless transmission line. The phasevelocity of the line is 0.7c, where c is the velocity of lightin a vacuum. Find v(z, t) and i(z, t) on the line.
24 Solution 5.6 Complete Solution for v(z, t) and i(z, t) We find the wavelength fromandThe voltage reflection coefficient at the load isThe input impedance of the line
25 Solution 5.6 Complete Solution for v(z, t) and i(z, t) Rewriting the expression for the generator voltage,Thus the phasor voltage isThe voltage on the line isand phasor voltage on the line is
26 Solution 5.6 Complete Solution for v(z, t) and i(z, t) The instantaneous voltage and current is
27 5-7 Special Cases of the Lossless Line Special cases has useful properties.For short-circuited line at z = −l,Short-Circuited Line
28 Example 5.7 Equivalent Reactive Elements Choose the length of a shorted 50- lossless transmission line (Fig. 5-16) such that its input impedance at 2.25 GHz is equivalent to the reactance of a capacitor with capacitance Ceq = 4 pF. The wave velocity on the line is 0.75c.
29 Solution 5.7 Equivalent Reactive Elements We are givenThe phase constant is2nd quadrant is4th quadrant isAny length l = 4.46 cm + nλ/2, where n is a positive integer, is also a solution.
30 Open-Circuited LineWith ZL = ∞, it forms an open-circuited line.
31 5-7.3 Application of Short-Circuit and Open-Circuit Measurements Product and ratio of SC and OC equations give the following results:Radio-frequency (RF) instruments measure the impedance of any load.
32 Example 5.8 Measuring Z0 and β Find Z0 and β of a 57-cm-long lossless transmissionline whose input impedance was measured as Zscin = j40.42Ω when terminated in a short circuit and as Zocin = −j121.24Ω when terminated in an open circuit. From other measurements, we know that the line is between 3 and 3.25 wavelengths long.
33 Solution 5.8 Measuring Z0 and β We have,True value of βl isand
34 Example 5.9 Quarter-Wave Transformer A 50-Ω lossless transmission line is to be matched to a resistive load impedance with ZL = 100Ω via a quarter-wave section as shown, thereby eliminating reflections along the feedline. Find the characteristic impedance of the quarter-wave transformer.
35 Solution 5.9 Quarter-Wave Transformer To eliminate reflections at terminal AA’, the input impedance Zin looking into the quarter-wave line should be equal to Z01, the characteristic impedance of the feedline. Thus, Zin = 50 .Since the lines are lossless, all the incident power will end up getting transferred into the load ZL.
36 5-8 Power Flow on a Lossless Transmission Line We shall examine the flow of power carried by incident and reflected waves.Instantaneous power is the product of instantaneous voltage and current.More interested in time-averaged power flow.Instantaneous PowerTime-Average Power
37 5-8.2 Time-Average Power There are 2 types of approach: 1) Time-Domain ApproachIncident power and reflected wave power areFor net average power delivered to the load,
38 5-8.2 Time-Average Power 5-9 Smith Chart 2) Phasor-Domain Approach Time-average power for any propagating wave isThe Smith Chart is used for analyzing and designing transmission-line circuits.5-9 Smith Chart
39 5-9 Smith Chart Impedances represented by normalized values, Z0. Reflection coefficient isNormalized load admittance is
40 Example 5.11 Determining ZL using the Smith Chart Given that the voltage standing-wave ratio is S = 3 ona 50-Ω line, that the first voltage minimum occurs at 5 cm from the load, and that the distance between successive minima is 20 cm, find the load impedance.SolutionThe first voltage minimum is at
41 Solution 5.11 Determining ZL using the Smith Chart From Smith Chart,The normalized loadimpedance at point C isMultiplying by Z0 = 50Ω ,we obtain
42 5-10 Impedance MatchingTransmission line is matched to the load when Z0 = ZL.Alternatively, place an impedance-matching network between load and transmission line.
43 Example 5.12 Single-Stub Matching 50-Ω transmission line is connected to an antennawith load impedance ZL = (25 − j50). Find the position and length of the short-circuited stub requiredto match the line.The normalized load impedance isLocated at point A.Solution
44 Solution 5.12 Single-Stub Matching Value of yL at B is which locates at position 0.115λ on the WTG scale.At C, located at 0.178λ on the WTG scale.Distant B and C isNormalized input admittance atthe juncture is
45 Solution 5.12 Single-Stub Matching Normalized admittance of −j 1.58 at F and position 0.34λ on the WTG scale givesAt point D,Distant B and C isNormalized input admittance at G.Rotating from point E to point G, we get