Transcript UNIT II
Transmission Line Concepts • • • • • • The transmission line is divided into small units where the circuit elements can be lumped. Assuming the resistance of the lines is zero, then the transmission line can be modeled as an LC ladder network with inductors in the series arms and the capacitors in the shunt arms. The value of inductance and capacitance of each part determines the velocity of propagation of energy down the line. Time taken for a wave to travel one unit length is equal to T(s) = (LC)0.5 Velocity of the wave is equal to v (m/s) = 1/T Impedance at any point is equal to Z = V (at any point)/I (at any point) Z = (L/C)0.5 1-5. Lossless transmission line R 0 and G 0 Propagation constant becomes k ZY j LC j Characteristic impedance becomes Z L Z0 Y C Voltage and current waves become jz V ( z) V e jz V e jz , I ( z) I e jz I e 2 General Input Impedance Equation • Input impedance of a transmission line at a distance L from the load impedance ZL with a characteristic Zo is Zinput = Zo [(ZL + j Zo BL)/(Zo + j ZL BL)] where B is called phase constant or wavelength constant and is defined by the equation B = 2p/l • Line terminated in its characteristic impedance: If the end of the transmission line is terminated in a resistor equal in value to the characteristic impedance of the line as calculated by the formula Z=(L/C)0.5 , then the voltage and current are compatible and no reflections occur. • Line terminated in a short: When the end of the transmission line is terminated in a short (RL = 0), the voltage at the short must be equal to the product of the current and the resistance. • Line terminated in an open: When the line is terminated in an open, the resistance between the open ends of the line must be infinite. Thus the current at the open end is zero. Reflections • In a line where the termination is equal to the impedance of the line, the reflections are zero • A line that is terminated other than Z0 is said to be mismatched and will have reflections • The reflection coefficient is found by: Vr Vi Wave Propagation on Lines • If a sine wave is applied to a transmission line, the signal moves down the line and disappears into the load • Such a signal is called a traveling wave • This process also takes time • A time delay of one period causes a phase shift of 360º, which is indistinguishable from the original • The length of a line L that causes a delay of one Traveling Waves Standing Waves • The interaction of incident and reflected waves in a transmission line results in standing waves • When a reflected wave is present but has lower amplitude than the incident, there will be no point on the line where the voltage or current remains zero over the whole cycle Variation of Impedance Along a Line • A matched line presents its impedance to a source located any distance from the load • An unmatched line impedance can vary greatly with its distance from the load • At some points mismatched lines may look inductive, other points may look capacitive, at still other points it may look resistive Impedance on a Lossless Line • The impedance on a lossless transmission line is given by the formula: Z L cos θ jZ0 sin θ Z Z0 Z 0 cos θ jZ L sin θ Characteristics of Open and Shorted Lines • An open or shorted line can be used as an inductive, capacitive, or even a resonant circuit • In practice, short-circuited sections are more common because open-circuited lines radiate energy from the open end • The impedance of a short-circuited line is: Z jZ0 tan θ Variation of Impedance Transmission Line Losses • No real transmission line is completely lossless • However, approximation is often valid assuming lossless lines Loss Mechanisms • The most obvious loss in a transmission line is due to the resistance of the line, called I2R loss • The dielectric can also cause loss, with the conductance becoming higher with increasing frequency • Open-wire systems can radiate energy – Loss becomes more significant as the frequency increases – Loss becomes worse as spacing between conductors increases Loss in Decibels • Transmission line losses are usually given in decibels per 100 feet or 100 meters • When selecting a transmission line, attention must be paid to the losses • A 3-dB loss equates to 1/2 the power being delivered to the antenna • Losses are also important in receivers where low noise depends upon minimizing the losses before the first stage of amplification Mismatched Lossy Lines • When a transmission line is lossy, the Standing- Wave Ratio (SWR) at the source is lower than that at the load • The reflection coefficient and standing-wave ratio both have larger magnitudes at the load • Computer programs and Smith Charts are available to calculate losses and mismatches in transmission lines Power Ratings • The maximum power that can be applied to a transmission line is limited by one of two things: – Power dissipation in the line – A maximum voltage, which can break down the dielectric when exceeded • A compromise is often achieved in power lines between voltage and line impedance