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**UNIVERSITI MALAYSIA PERLIS**

EKT 241/4: ELECTROMAGNETIC THEORY UNIVERSITI MALAYSIA PERLIS CHAPTER 5 – TRANSMISSION LINES PREPARED BY: NORDIANA MOHAMAD SAAID

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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

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**General Considerations**

Transmission line – a two-port network connecting a generator circuit to a load.

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The role of wavelength The impact of a transmission line on the current and voltage in the circuit depends on the length of line, l and the frequency, f of the signal provided by generator. At low frequency, the impact is negligible At high frequency, the impact is very significant

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**Propagation modes Transmission lines may be classified into two types:**

a) Transverse electromagnetic (TEM) transmission lines – waves propagating along these lines having electric and magnetic field that are entirely transverse to the direction of propagation b) Higher order transmission lines – waves propagating along these lines have at least one significant field component in the direction of propagation

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Propagation modes

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Lumped- element model

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Lumped- element model A transmission line is represented by a parallel-wire configuration regardless of the specific shape of the line, i.e coaxial line, two-wire line or any TEM line. Lumped element circuit model consists of four basic elements called ‘the transmission line parameters’ : R’ , L’ , G’ , C’ .

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**Lumped- element model Lumped-element transmission line parameters:**

R’ : combined resistance of both conductors per unit length, in Ω/m L’ : the combined inductance of both conductors per unit length, in H/m G’ : the conductance of the insulation medium per unit length, in S/m C’ : the capacitance of the two conductors per unit length, in F/m

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Lumped- element model Note: µ, σ, ε pertain to the insulating material between conductors

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Lumped- element model All TEM transmission lines share the following relation: Note: µ, σ, ε pertain to the insulating material between conductors

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**Transmission line equations**

Complex propagation constant, γ α – the real part of γ - attenuation constant, unit: Np/m β – the imaginary part of γ - phase constant, unit: rad/m

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**Transmission line equations**

The characteristic impedance of the line, Z0 : Phase velocity of propagating waves: where f = frequency (Hz) λ = wavelength (m)

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Example 1 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 50Ω and phase constant of 20 rad/m at 700MHz, find the inductance per meter and the capacitance per meter of the line.

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**Solution to Example 1 The following quantities are given:**

With R’ = G’ = 0, The ratio is given by We get L’ from Z0

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**Lossless transmission line**

Lossless transmission line - Very small values of R’ and G’. We set R’=0 and G’=0, hence:

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**Lossless transmission line**

Using the relation properties between μ, σ, ε : Wavelength, λ Where εr = relative permittivity of the insulating material between conductors

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**Voltage reflection coefficient**

The load impedance, ZL Where; = total voltage at the load V0- = amplitude of reflected voltage wave V0+ = amplitude of the incident voltage wave = total current at the load Z0 = characteristic impedance of the line

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**Voltage reflection coefficient**

Hence, load impedance, ZL: Solving in terms of V0- :

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**Voltage reflection coefficient**

Voltage reflection coefficient, Γ – the ratio of the amplitude of the reflected voltage wave, V0- to the amplitude of the incident voltage wave, V0+ at the load. Hence,

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**Voltage reflection coefficient**

Z0 for lossless line is a real number while ZL in general is a complex number. Hence, Where |Γ| = magnitude of Γ θr = phase angle of Γ A load is matched to the line if ZL = Z0 because there will be no reflection by the load (Γ = 0 and V0−= 0.

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**Voltage reflection coefficient**

When the load is an open circuit, (ZL=∞), Γ = 1 and V0- = V0+. When the load is a short circuit (ZL=0), Γ = -1 and V0- = V0+.

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Example 2 A 100-Ω transmission line is connected to a load consisting of a 50-Ω resistor in series with a 10pF capacitor. Find the reflection coefficient at the load for a 100-MHz signal.

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**Solution to Example 2 The following quantities are given**

The load impedance is Voltage reflection coefficient is

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Standing Waves Interference of the reflected wave and the incident wave along a transmission line creates a standing wave. Constructive interference gives maximum value for standing wave pattern, while destructive interference gives minimum value. The repetition period is λ for incident and reflected wave individually. But, the repetition period for standing wave pattern is λ/2.

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**Standing Waves For a matched line, ZL = Z0, Γ = 0 and**

= |V0+| for all values of z.

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Standing Waves For a short-circuited load, (ZL=0), Γ = -1.

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**Standing Waves For an open-circuited load, (ZL=∞), Γ = 1.**

The wave is shifted by λ/4 from short-circuit case.

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**Standing Waves First voltage maximum occurs at:**

First voltage minimum occurs at: For next voltage maximum: Where θr = phase angle of Γ

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**Voltage standing wave ratio**

VSWR is the ratio of the maximum voltage amplitude to the minimum voltage amplitude: VSWR provides a measure of mismatch between the load and the transmission line. For a matched load with Γ = 0, VSWR = 1 and for a line with |Γ| - 1, VSWR = ∞.

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Example 3 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 (VSWR).

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Solution to Example 3 We have, VSWR is given by:

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**Input impedance of a lossless line**

The input impedance, Zin is the ratio of the total voltage (incident and reflected voltages) to the total current at any point z on the line. or

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**Special cases of the lossless line**

For a line terminated in a short-circuit, ZL = 0: For a line terminated in an open circuit, ZL = ∞:

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**Application of short-circuit and open-circuit measurements**

The measurements of short-circuit input impedance, and open-circuit input impedance, can be used to measure the characteristic impedance of the line: and

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Length of line If the transmission line has length , where n is an integer, Hence, the input impedance becomes:

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**Quarter wave transformer**

If the transmission line is a quarter wavelength, with , where , we have , then the input impedance becomes:

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Example 4 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 to Example 4 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 = 50Ω . Since the lines are lossless, all the incident power will end up getting transferred into the load ZL.

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**Matched transmission line**

For a matched lossless transmission line, ZL=Z0: 1) The input impedance Zin=Z0 for all locations z on the line, 2) Γ =0, and 3) all the incident power is delivered to the load, regardless of the length of the line, l.

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**Power flow on a lossless transmission line**

Two ways to determine the average power of an incident wave and the reflected wave; Time-domain approach Phasor domain approach Average power for incident wave; Average power for reflected wave:

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**Power flow on a lossless transmission line**

The net average power delivered to the load:

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Smith Chart Smith chart is used to analyze & design transmission line circuits. Impedances on Smith chart are represented by normalized value, zL for example: the normalized load impedance, zL is dimensionless.

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**Smith Chart Reflection coefficient, Γ : Since , Γ becomes:**

Re-arrange in terms of zL: Normalized load admittance:

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Smith Chart The complex Γ plane.

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Smith Chart The families of circle for rL and xL.

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Smith Chart Plotting normalized impedance, zL = 2-j1

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**Input impedance The input impedance, Zin:**

Γ is the voltage reflection coefficient at the load. We shift the phase angle of Γ by 2βl, to get ΓL. This will zL to zin. The |Γ| is the same, but the phase is changed by 2βl. On the Smith chart, this means rotating in a clockwise direction (WTG).

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Input impedance Since β = 2π/λ, shifting by 2 βl is equal to phase change of 2π. Equating: Hence, for one complete rotation corresponds to l = λ/2. The objective of shifting Γ to ΓL is to find Zin at an any distance l on the transmission line.

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Example 5 A 50-Ω transmission line is terminated with ZL=(100-j50)Ω. Find Zin at a distance l =0.1λ from the load.

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Solution to Example 5 at B, zin = 0.6 –j0.66

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**VSWR, Voltage Maxima and Voltage Minima**

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**VSWR, Voltage Maxima and Voltage Minima**

Point A is the normalized load impedance with zL=2+j1. VSWR = 2.6 (at Pmax). The distance between the load and the first voltage maximum is lmax=( )λ. The distance between the load and the first voltage minimum is lmin=( )λ.

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**Impedance to admittance transformations**

zL=0.6 + j1.4 yL= j0.6

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Example 6 Given that the voltage standing-wave ratio, VSWR = 3. On a 50-Ω line, 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.

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Solution to Example 6 The distance between successive minima is equal to λ/2. Hence, λ = 40 cm. First voltage minimum (in wavelength unit) is at on the WTL scale from point B. Intersect the line with constant SWR circle = 3. The normalized load impedance at point C is: De-normalize (multiplying by Z0) to get ZL:

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Solution to Example 6

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Impedance Matching Transmission line is matched to the load when Z0 = ZL. This is usually not possible since ZL is used to serve other application. Alternatively, we can place an impedance-matching network between load and transmission line.

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Single- stub matching Matching network consists of two sections of transmission lines. First section of length d, while the second section of length l in paralllel with the first section, hence it is called stub. The second section is terminated with either short-circuit or open circuit.

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Single- stub matching

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Single- stub matching The distance d is chosen so as to transform the load admittance, YL=1/ZL into an admittance of the form Yd = Y0+jB when looking towards the load at MM’. The length l of the stub is chosen so that its input admittance, YS at MM’ is equal to –jB. Hence, the parallel sum of the two admittances at MM’ yields Y0, which is the characteristic admittance of the line.

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Example 7 50-Ω transmission line is connected to an antenna with load impedance ZL = (25 − j50)Ω. Find the position and length of the short-circuited stub required to match the line.

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**Solution to Example 7 The normalized load impedance is (located at A).**

Value of yL at B is which locates at position 0.115λ on the WTG scale. Draw constant SWR circle that goes through points A and B. There are two possible matching points, C and D where the constant SWR circle intersects with circle rL=1 (now gL =1 circle).

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Solution to Example 7

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**Solution to Example 7 First matching points, C.**

At C, is at 0.178λ on WTG scale. Distance B and C is Normalized input admittance at the juncture is: E is the admittance of short-circuit stub, yL=-j∞. Normalized admittance of −j 1.58 at F and position 0.34λ on the WTG scale gives:

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**Solution to Example 7 Second matching point, D. At point D,**

Distance B and C is Normalized input admittance at G. Rotating from point E to point G, we get

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Solution to Example 7

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