Chapter 4 1 Transmission Lines A transmission line

Chapter 4 1

Transmission Lines A transmission line connects a generator to a load Transmission lines include: • Two parallel wires • Coaxial cable • Microstrip line • Optical fiber • Waveguide • etc.

Example of TEM Mode Electric Field E is radial Magnetic Field H is azimuthal Propagation is into the page

Transmission line equations Telegraphist’s Equation is used to determine the basic characteristics for transmission line, which are: 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, Z 0 : Phase velocity of propagating waves: where f = frequency (Hz) λ = wavelength (m)

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

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

Lumped-element transmission line parameters:

Lumped- element model

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)

5 -2 Lumped-Element Model • In summary, • All TEM transmission lines share the relations:

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

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

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)

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.

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

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

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.

5 -5. 1 Voltage Reflection Coefficient • The relations for lossless are • A load that is matched to the line when ZL = Z 0, Γ = 0 and V 0−= 0.

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 -p. F capacitor. Find the reflection coefficient at the load for a 100 -MHz signal.

Solution 5. 2 Reflection Coefficient of a Series RC Load The following quantities are given The load impedance is Voltage reflection coefficient is

Smith Chart • Smith chart is used to analyze & design transmission line circuits. • Reduce tedious work of complex mathematics in high frequency transmission line problems. • Plotted on the complex reflection coefficient plane in two dimensions real and imaginary(normalized impedance). • Impedances on Smith chart are represented by normalized value, z. L for example:

5 -9 Smith Chart • Impedances represented by normalized values, Z 0. • Reflection coefficient is • Normalized load admittance is

Voltage reflection coefficient • Voltage reflection coefficient, Γ – the ratio of the amplitude of the reflected voltage wave, V 0 - to the amplitude of the incident voltage wave, V 0+ at the load. • Hence,

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 = Z 0, Γ = 0 and = |V 0+| 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.

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 = ∞.

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