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EKT241 – ELECTROMAGNETICS THEORY Chapter 5 Transmission Lines

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Chapter Objectives Introduction to transmission lines Lump-element model that represent TEM lines Lossless line Smith Chart to analyze transmission line problem

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Chapter Outline General Considerations Lumped-Element Model Transmission-Line Equations Wave Propagation on a Transmission Line The Lossless Transmission Line Input Impedance of the Lossless Line Special Cases of the Lossless Line Power Flow on a Lossless Transmission Line The Smith Chart Impedance Matching Transients on Transmission Lines 5-1) 5-2) 5-3) 5-4) 5-5) 5-6) 5-7) 5-8) 5-9) 5-10) 5-11)

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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.

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5-2 Lumped-Element Model Transmission lines can be represented by a lumped-element circuit model.

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5-2 Lumped-Element Model Lumped-element circuit model consists 4 transmission line parameters: 1.R’ (Ω/m) 2.L’ (H/m) 3.G’ (S/m) 4.C’ (F/m)

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5-2 Lumped-Element Model In summary, All TEM transmission lines share the relations: where µ, σ, ε = properties of conductor

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5-3 Transmission-Line Equations Transmission line equations in phasor form is given as

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5-4 Wave Propagation on a Transmission Line The wave equation is derived as γ has real part α (attenuation constant) and imaginary part β (phase constant). Complex propagation constant

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5-4 Wave Propagation on a Transmission Line Characteristic impedance Z 0 of the line is Phase velocity for propagating wave is where f = frequency (Hz) λ = wavelength (m)

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Example 5.1 Air Line An air line is a transmission line for which air is the dielectric 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.

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Solution 5.1 Air Line The following quantities are given: With R’ = G’ = 0, The ratio is given by We get L’ from Z 0

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5-5 The Lossless Transmission Line Low R’ and G’ for transmission line is called lossless transmission line. Using relation properties,

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5-5 The Lossless Transmission Line Wavelength is given by where ε r = relative permittivity For the lossless line, there are 2 unknowns in the equations for the total voltage and current on the line.

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5-5.1 Voltage Reflection Coefficient The relations for lossless are A load that is matched to the line when Z L = Z 0, Γ = 0 and V 0 − = 0.

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Example 5.2 Reflection Coefficient of a Series RC Load A 100-Ω transmission line is connected to a load consisting of a 50-Ω resistor in series with a 10-pF capacitor. Find the reflection coefficient at the load for a 100-MHz signal.

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Solution 5.2 Reflection Coefficient of a Series RC Load The following quantities are given The load impedance is Voltage reflection coefficient is

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5-5.2 Standing Waves 3 types of voltage standing-wave patterns: (a) Matched load (b) Short-circuited line (c) Open-circuited line

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5-5.2 Standing Waves To find maximum and minimum values of voltage magnitude, we have

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5-5.2 Standing Waves First voltage maximum occurs at First voltage minimum occurs at Voltage standing-wave ratio S is defined as

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Example 5.4 Standing-wave Ratio A 50- transmission line is terminated in a load with Z L = (100 + j50)Ω. Find the voltage reflection coefficient and the voltage standing-wave ratio (SWR). We have, S is given by Solution

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5-6 Input Impedance of the Lossless Line Voltage to current ratio is called input impedance Z in. The input impedance at z = −l is given as and

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Example 5.6 Complete Solution for v(z, t) and i(z, t) A 1.05-GHz generator circuit with series impedance Z g = 10Ω and voltage source given by is connected to a load Z L = (100 + j50) through a 50-Ω, 67-cm-long lossless transmission line. The phase velocity of the line is 0.7c, where c is the velocity of light in a vacuum. Find v(z, t) and i(z, t) on the line.

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Solution 5.6 Complete Solution for v(z, t) and i(z, t) We find the wavelength from and The voltage reflection coefficient at the load is The input impedance of the line

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Solution 5.6 Complete Solution for v(z, t) and i(z, t) Rewriting the expression for the generator voltage, Thus the phasor voltage is The voltage on the line is and phasor voltage on the line is

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Solution 5.6 Complete Solution for v(z, t) and i(z, t) The instantaneous voltage and current is

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5-7 Special Cases of the Lossless Line Special cases has useful properties. For short-circuited line at z = −l, Short-Circuited Line

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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 C eq = 4 pF. The wave velocity on the line is 0.75c.

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Solution 5.7 Equivalent Reactive Elements We are given The phase constant is 2 nd quadrant is 4 th quadrant is Any length l = 4.46 cm + nλ/2, where n is a positive integer, is also a solution.

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5-7.2 Open-Circuited Line With Z L = ∞, it forms an open-circuited line.

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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.

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Example 5.8 Measuring Z 0 and β Find Z 0 and β of a 57-cm-long lossless transmission line whose input impedance was measured as Z sc in = j40.42Ω when terminated in a short circuit and as Z oc in = −j121.24Ω when terminated in an open circuit. From other measurements, we know that the line is between 3 and 3.25 wavelengths long.

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Solution 5.8 Measuring Z 0 and β We have, True value of βl is and

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Example 5.9 Quarter-Wave Transformer A 50-Ω lossless transmission line is to be matched to a resistive load impedance with Z L = 100Ω via a quarter- wave section as shown, thereby eliminating reflections along the feedline. Find the characteristic impedance of the quarter-wave transformer.

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Solution 5.9 Quarter-Wave Transformer To eliminate reflections at terminal AA’, the input impedance Z in looking into the quarter-wave line should be equal to Z 01, the characteristic impedance of the feedline. Thus, Z in = 50. Since the lines are lossless, all the incident power will end up getting transferred into the load Z L.

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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 Power Time-Average Power

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There are 2 types of approach: 1) Time-Domain Approach Incident power and reflected wave power are For net average power delivered to the load,

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5-8.2 Time-Average Power 2) Phasor-Domain Approach Time-average power for any propagating wave is The Smith Chart is used for analyzing and designing transmission-line circuits. 5-9 Smith Chart

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Impedances represented by normalized values, Z 0. Reflection coefficient is Normalized load admittance is

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Example 5.11 Determining Z L using the Smith Chart Given that the voltage standing-wave ratio is S = 3 on a 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. Solution The first voltage minimum is at

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Solution 5.11 Determining Z L using the Smith Chart From Smith Chart, The normalized load impedance at point C is Multiplying by Z 0 = 50Ω, we obtain

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5-10 Impedance Matching Transmission line is matched to the load when Z 0 = Z L. Alternatively, place an impedance-matching network between load and transmission line.

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Example 5.12 Single-Stub Matching 50-Ω transmission line is connected to an antenna with load impedance Z L = (25 − j50). Find the position and length of the short-circuited stub required to match the line. The normalized load impedance is Located at point A. Solution

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Solution 5.12 Single-Stub Matching Value of y L 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 is Normalized input admittance at the juncture is

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Solution 5.12 Single-Stub Matching Normalized admittance of −j 1.58 at F and position 0.34λ on the WTG scale gives At point D, Distant B and C is Normalized input admittance at G. Rotating from point E to point G, we get

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Solution 5.12 Single-Stub Matching

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5-11 Transients on Transmission Lines Transient response is a time record of voltage pulse. An example of step function is shown below.

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Transient Response Steady-state voltage V ∞ for d-c analysis of the circuit is where V g = DC voltage source Steady-state current is

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