EKT 241/4: ELECTROMAGNETIC THEORY

EKT 241/4:ELECTROMAGNETIC THEORY UNIVERSITI MALAYSIA PERLIS CHAPTER 5 – TRANSMISSION LINES PREPARED BY: SaidatulNorlyanaAzemi snorlyana@unimap.edu.my

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

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

So…What is the use of transmission line?? • A transmission line is used to transmit electrical energy/signals from one point to another • i.e. from one source to a load. • Types of transmission line include: wires, (telephone wire), coaxial cables, optical fibers n etc…

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

Propagation modes Electric field lines Magnetic field lines

Propagation modes A few examples of transverse electromagnetic (TEM) and higher order transmission line

Lumped- element model • A transmission line is represented by a parallel-wire configuration regardless of the specific shape of the line, (in term of lumped element circuit model) • 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’ . Series element Shunt element

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 • For example, a coil of wire has the property of inductance. When a certain amount of inductance is needed in a circuit, a coil of the proper dimension is inserted

Lumped- element model

Lumped- element model for 3 type of lines Note: µ, σ, ε pertain to the insulating material between conductors

Exercise 1: • Use table 5.1 to compute the line parameter of a two wire air line whose wires are separated by distance of 2 cm, and, each is 1 mm in radius. The wires may be treated as perfect conductors with σc=. R’ = ?, L’=?, G’=?, C’=?

Solution exercise 1: σc= σc=

Exercise 2: • Calculate the transmission line parameters at 1 MHz for a rigid coaxial air line with an inner conductor diameter of 0.6 cm and outer conductor diameter of 1.2 cm. The conductors are made of copper. (μc=0.9991 ; σc=5.8x107) f = 1MHz r1 = 0.006m/2 = 0.003m r2 = 0.012m/2 = 0.006m

Solution exercise 2:

BARE IN UR MIND From calculator

Because, the material separating the inner and outer is perfect dielectric (air) with σ=0, thus G’ = 0 • G’ : the conductance of the insulation medium per unit length, in S/m

Transmission line equations Is used to describes the voltage and the current across the transmission line in term of propagation constant and impedance • 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

Transmission line equations • The characteristic impedance of the line, Z0: • Phase velocity of propagating waves: where f = frequency (Hz) λ = wavelength (m) β= phase constant

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

Solution to Example 1 • The following quantities are given: • WithR’ = G’ = 0,

Solution to Example 1 2 • The ratio is given by: • We get L’ from Z0

Lossless transmission line Transmission line can be designed to minimize ohmic losses by selecting high conductivities and dielectric material, thus we assume : • Lossless transmission line - Very small values of R’ and G’. • We set R’=0 and G’=0, hence:

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 0 0

Lossless transmission line Transmission line can be designed to minimize ohmic losses by selecting high conductivities and dielectric material, thus we assume : • Lossless transmission line - Very small values of R’ and G’. • We set R’=0 and G’=0, hence:

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

Exercise 3: • For a losses transmission line, λ = 20.7 cm at 1GHz. Find εrof the insulating material. λ=20.7cm 0.207m ; f=1 GHz 2

Exercise 4 • A lossless transmission line of length 80 cm operates at a frequency of . The line parameters are & Find the characteristic impedance, the phase constant and the phase velocity. The condition apply that the line is lossless, So: R= 0 & G=0

characteristic impedance : • phase constant: With R n G = 0 = 18.85 rad/m

phase velocity:

Voltage Reflection Coefficient • Every transmission line has a resistance associated with it, and comes about because of its construction. This is called its characteristic impedance, Z0. • The standard characteristic impedance value is 50Ω. However when the transmission line is terminated with an arbitrary load ZL, in which is not equivalent to its characteristic impedance (ZL ≠ Z0), a reflected wave will occur.

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,

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

Voltage reflection coefficient • And in case of a RL and RC series, ZL: ZL = R + jL ; ZL = R -1/jC • A load is matched to the line if ZL = Z0 because there will be no reflection by the load (Γ= 0and V0−= 0. • 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+.

What is the difference between an open and closed circuit? • closed allows electricity through, and open doesn't. • open circuit - Any circuit which is not complete is considered an open circuit. The open status of the circuit doesn't depend on how it became unclosed, so circuits which are manually disconnected and circuits which have blown fuses, faulty wiring or missing components are all considered open circuits. • close circuit: A circuit is considered to be closed when electricity flows from an energy source to the desired endpoint of the circuit. A complete circuit which is not performing any actual work can still be a closed circuit. For example, a circuit connected to a dead battery may not perform any work, but it is still a closed circuit.

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

Solution to Example 2 • The following quantities are given • The load impedance is • Voltage reflection coefficient is

In order to convert from –ve magnitude for Г by replacing the –ve sign with e-j180

Math’s TIP… 1 2

Exercise 5 • A 150 Ωlossless line is terminated in a load impedance ZL= (30 –j200) Ω. Calculate the voltage reflection coefficient at the load. Zo = 150 Ω ZL= (30 –j200) Ω

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.

Standing Waves • For a matched line, ZL = Z0, Γ= 0and = |V0+| for all values of z.

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

Standing Waves • For an open-circuited load, (ZL=∞), Γ = 1. The wave is shifted by λ/4 from short-circuit case.

Standing Waves • First voltage maximum occurs at: • If θr ≥ 0  n=0; • If θr ≤ 0  n=1 • First voltage minimumoccurs at: Where θr = phase angle of Γ

VSWR • Voltage Standing Wave Ratio (VSWR) is ratio between the maximum voltage an the minimum voltage along the transmission line. • 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 = ∞. The VSWR is given by:

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).

EKT 241/4: ELECTROMAGNETIC THEORY