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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 5-1) 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 5-2) 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: R’ (Ω/m) L’ (H/m) G’ (S/m) 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 Z0 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 Z0

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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 εr = 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 ZL = Z0, Γ = 0 and V0−= 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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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 ZL = (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 Zin. 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 Zg = 10Ω and voltage source given by is connected to a load ZL = (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 Ceq = 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 2nd quadrant is 4th quadrant is Any length l = 4.46 cm + nλ/2, where n is a positive integer, is also a solution.

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Open-Circuited Line With ZL = ∞, 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 Z0 and β**

Find Z0 and β of a 57-cm-long lossless transmission line 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.

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**Solution 5.8 Measuring Z0 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 ZL = 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 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.

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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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**5-8.2 Time-Average Power 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 5-9 Smith Chart 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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**5-9 Smith Chart Impedances represented by normalized values, Z0.**

Reflection coefficient is Normalized load admittance is

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**Example 5.11 Determining ZL 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 ZL using the Smith Chart**

From Smith Chart, The normalized load impedance at point C is Multiplying by Z0 = 50Ω , we obtain

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5-10 Impedance Matching Transmission line is matched to the load when Z0 = ZL. 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 ZL = (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 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 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 Vg = DC voltage source Steady-state current is

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