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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: [email protected] Antennas: from Theory to Practice 1 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. Antennas: from Theory to Practice 2 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. Antennas: from Theory to Practice 3 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. Antennas: from Theory to Practice 4 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. Antennas: from Theory to Practice 5 Transmission Line Model A distributed element system is converted to a lumped one Antennas: from Theory to Practice 6 Antennas: from Theory to Practice 7 Transmission line equation Where the propagation constant: Attenuation const: Phase const: Antennas: from Theory to Practice 8 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 Antennas: from Theory to Practice 9 Forward and reverse travelling waves Velocity: , so it is also called the wave number Antennas: from Theory to Practice 10 Lossless transmission lines • For a lossless transmission line, R = G =0, Antennas: from Theory to Practice 11 Terminated Transmission Line • Input impedance and reflection coefficient Antennas: from Theory to Practice 12 Note: the power reflection coefficient is: The input impedance For the lossless case Antennas: from Theory to Practice 13 Input impedance for special cases • Matched case (G = 0): • Open circuit (G = 1): • Short circuit (G = -1): • Quarter-wavelength case: Antennas: from Theory to Practice 14 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! Antennas: from Theory to Practice 15 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 Antennas: from Theory to Practice 16 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. Antennas: from Theory to Practice 17 Antennas: from Theory to Practice 18 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 Antennas: from Theory to Practice 19 Antennas: from Theory to Practice 20 2.3 The Smith Chart and Impedance Matching Antennas: from Theory to Practice 21 The Smith Chart Antennas: from Theory to Practice 22 Example 2.7 Using a Smith Chart to redo Example 2.1, and also display the reflection coefficient on the Chart. Antennas: from Theory to Practice 23 Impedance Matching • 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. Antennas: from Theory to Practice 24 Antennas: from Theory to Practice 25 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 Antennas: from Theory to Practice 26 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. Antennas: from Theory to Practice 27 Distributed matching networks They can be formed by a quarter-wavelength transmission line, an open-circuit/short-circuit transmission line, or their combinations. Antennas: from Theory to Practice 28 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. Antennas: from Theory to Practice 29 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 Antennas: from Theory to Practice 30 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 Antennas: from Theory to Practice 31 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. 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. • Antennas are designed to have a low Q, whereas circuit components are designed for a high Q. Antennas: from Theory to Practice 32 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. Antennas: from Theory to Practice 33 2.4 Various Transmission Lines Antennas: from Theory to Practice 34 Two-wire Transmission Line • Characteristic impedance (for lossless line): Typical value is 300 Ω Antennas: from Theory to Practice 35 • 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 electromagnetic) 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. Antennas: from Theory to Practice 36 Coaxial Cable Velocity in a medium Antennas: from Theory to Practice 37 • Fundamental mode: – TEM mode below the cut-off freq • Characteristic impedance: The typical value for industrial standard lines is 50 Ω or 75 Ω, do you know why? • Loss Antennas: from Theory to Practice 38 Cable examples Antennas: from Theory to Practice 39 Microstrip Line Effective relative permittivity: thus - determined by the capacity Antennas: from Theory to Practice 40 • Characteristic impedance: , W/d <1 , W/d >1 • Basic mode: quasi-TEM mode if the wavelength larger than the cut-off wavelength: Antennas: from Theory to Practice 41 • Loss • Surface waves and cut-off frequencies Antennas: from Theory to Practice 42 Antennas: from Theory to Practice 43 Stripline • Characteristic impedance Antennas: from Theory to Practice 44 • Fundamental mode: TEM mode if • Loss – Similar to that of microstrip, but little radiation loss and surface wave loss. Antennas: from Theory to Practice 45 Co-planar Waveguide (CPW) where Antennas: from Theory to Practice 46 • Characteristic impedance where Antennas: from Theory to Practice 47 • Fundamental mode: quasi-TEM mode • Loss – Normally higher than microstrip Antennas: from Theory to Practice 48 Waveguides • There are circular and rectangular waveguides which have just one piece of conductor, and good for high frequencies (high pass, and low stop). Antennas: from Theory to Practice 49 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 Antennas: from Theory to Practice 50 • Fundamental mode: TE10 mode Thus its cut-off wavelength is 2a, and the operational wavelength should shorter than 2a. Antennas: from Theory to Practice 51 • Waveguide wavelength: the period of the wave inside the waveguide. • Characteristic impedance Antennas: from Theory to Practice 52 Comparison of transmission lines Antennas: from Theory to Practice 53 2.5 Connectors Male (left) and female (right) N-type connectors Antennas: from Theory to Practice 54 Antennas: from Theory to Practice 55