Yi HUANG Department of Electrical Engineering & Electronics Antennas: from Theory to Practice 2. Circuit Concepts and Transmission Lines Yi HUANG Department of Electrical Engineering & Electronics The University of Liverpool Liverpool L69 3GJ Email: Yi.Huang@liv.ac.uk This is a general rule, but it may need modification for certain situation.

Objectives of This Chapter Review the very basics of circuit concepts; Distinguish the lumped element system from the distributed element system; Introduce the fundamentals of transmission lines; Compare various transmission lines and connectors.

2.1 Circuit Concepts Electric current I is a measure of the charge flow/ movement. Voltage V is the difference of electrical potential between two points of an electrical or electronic circuit. Impedance Z = R + jX is a measure of opposition to an electric current.

Lumped and Distributed Element Systems The current and voltage along a transmission line may be considered unchanged (which normally means the frequency is very low). The system is called a lumped element system. The current and voltage along a transmission line are functions of the distance from the source (which normally means the frequency is high), thus the system is called a distributed element system.

2.2 Transmission Line Theory A transmission line is the structure that forms all or part of a path from one place to another for directing the transmission of energy, such as electrical power transmission and microwaves. We are only interested in the transmission lines for RF engineering and antenna applications. Thus dielectric transmission lines such as optical fibres are not considered.

Transmission Line Model A distributed element system is converted to a lumped one

Transmission line equation Where the propagation constant: Attenuation const: Phase const:

The solutions are: This is the characteristic impedance of the transmission line. For a lossless transmission line, R = G =0, thus The industrial standard transmission line normally has a characteristic impedance of 50 or 75 Ω

Forward and reverse travelling waves Velocity: , so it is also called the wave number

Lossless transmission lines For a lossless transmission line, R = G =0,

Terminated Transmission Line Input impedance and reflection coefficient

Note: the power reflection coefficient is: The input impedance For the lossless case

Input impedance for special cases Matched case (G = 0): Open circuit (G = 1): Short circuit (G = -1): Quarter-wavelength case:

Input impedance for ZL = 75  and Z0 = 50  - a period function! Example 2.1 A lossless transmission line with a characteristic impedance of 50  is loaded by a 75  resistor. Plot the input impedance as a function of the line length (up to two wavelengths). Input impedance for ZL = 75  and Z0 = 50  - a period function!

Return loss When the voltage reflection coefficient and power reflection coefficient are expressed in logarithmic forms, they give the same result, which is called the return loss

Example 2.5 A 75  resistor is connected to a low loss transmission line with characteristic impedance of 50 . The attenuation constant is 0.2 Np/m at 1 GHz. a). What is the voltage reflection coefficient for l = 0 and l/4, respectively? b). Plot the return loss as a function of the line length. Assume that the effective relative permittivity is 1.5.

Voltage Standing Wave Ratio (VSWR) The VSWR (also known as the standing wave ratio, SWR) is defined as the magnitude ratio of the maximum voltage on the line to the minimum voltage on the line

2.3 The Smith Chart and Impedance Matching

The Smith Chart

Example 2.7 Using a Smith Chart to redo Example 2.1, and also display the reflection coefficient on the Chart.

Impedance Matching Ideally: Impedance matching is the practice of making the output impedance of a source equal to the input impedance of the load in order to maximize the power transfer and minimize reflections from the load. Mathematically, it means the load impedance being the complex conjugate of the source impedance. Ideally: Generally speaking, resistors are not employed for impedance matching The lumped matching networks can be divided into three basic networks: L network, T network and pi () network.

Lumped T and P networks T network which may be viewed as another reactance (jX2) added to the L network P network can be seen as an admittance (jB2) added to the L network

Example 2.8 A load with an impedance of 10-j100  is to be matched with a 50  transmission line. Design a matching network and discuss if there are other solutions available.

Distributed matching networks They can be formed by a quarter-wavelength transmission line, an open-circuit/short-circuit transmission line, or their combinations.

Example 2.9 A load with an impedance of 10-j100  is to be matched with a 50  transmission line. Design two distributed matching networks and compare them in terms of the bandwidth performance.

A). a short circuit with a stub length l2 = 0.0325l; B). an open circuit with a stub length l2 = 0.2825l. Both have achieved a perfect matching at 1GHz but of different bandwidth

Frequency bandwidth limitation There exists a general limit on the bandwidth over which an arbitrarily good impedance match can be obtained in the case of a complex load impedance. It is related to the ratio of reactance to resistance, and to the bandwidth over which we desire to match the load. Take the parallel RC load impedance as an example, Bode and Fano derived, for lumped circuits, a fundamental limitation for it and it can be expressed as

Quality Factor and Bandwidth Quality factor, Q, which is a measure of how much lossless reactive energy is stored in a circuit compared to the average power dissipated. Antennas are designed to have a low Q, whereas circuit components are designed for a high Q. where WE is the energy stored in the electric field, WM is the energy stored in the magnetic field and PL is the average power delivered to the load.

where f1 and f2 are the frequencies at which the power reduces to half of its maximum value at the resonant frequency, f0 and where BF is the fractional bandwidth. This relation only truly applies to simple (unloaded single resonant) circuits.

2.4 Various Transmission Lines

Two-wire Transmission Line Characteristic impedance (for lossless line): Typical value is 300 Ω

Twisted-pair transmission line Fundamental mode Both the electric field and magnetic field are within the transverse (to the propagation direction) plane, thus this mode is called the TEM (transverse electro-magnetic) mode. Loss the principle loss is actually due to radiation, especially at higher frequencies. The typical usable frequency is less than 300 MHz Twisted-pair transmission line the twisted configuration has cancelled out the radiation from both wires and resulted in a small and symmetrical total field around the line; but it is not suitable for high frequencies due to the high dielectric losses that occur in the insulation.

Coaxial Cable Velocity in a medium

Characteristic impedance: Fundamental mode: TEM mode below the cut-off freq Characteristic impedance: Loss The typical value for industrial standard lines is 50 Ω or 75 Ω, do you know why?

Cable examples

Microstrip Line Effective relative permittivity: thus - determined by the capacity

Characteristic impedance: Basic mode: quasi-TEM mode if the wavelength larger than the cut-off wavelength: , W/d <1 , W/d >1

Loss Surface waves and cut-off frequencies

Stripline Characteristic impedance

Fundamental mode: TEM mode if Loss Similar to that of microstrip, but little radiation loss and surface wave loss.

Co-planar Waveguide (CPW) where

Characteristic impedance where

Fundamental mode: quasi-TEM mode Loss Normally higher than microstrip

Waveguides There are circular and rectangular waveguides which have just one piece of conductor, and good for high frequencies (high pass, and low stop).

Standard waveguides The frequency range is determined by the cut-off frequencies of the fundamental mode and the 1st higher mode. The cut-off wavelength for TEmn and TMmn modes is given by

Fundamental mode: TE10 mode Thus its cut-off wavelength is 2a, and the operational wavelength should shorter than 2a.

Waveguide wavelength: the period of the wave inside the waveguide. Characteristic impedance

Comparison of transmission lines

2.5 Connectors Male (left) and female (right) N-type connectors