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2.4 Finite Length transmission line 43 2.4 Finite Length transmission line We have determined that, if we had a transmission line of infinite length (Figure 2.4-1), we would be able to measure a finite resistance between inner and outer conductor (usually 50 Ω) which is termed characteristic impedance of the line. But in reality, since the line cannot be infinite in length, we would always see an infinite resistance between our two wires. Nonetheless, the characteristic impedance rating of a transmission line is important even when dealing with limited lengths. If we had a finite length of line which was left open-circuited at the end, we would initially observe the distributed inductance and capacitance progressively charging just as we had in the case an an infinitely long line (Figure 2.3-6). From the point of a view of the battery, up until the time it takes for the unterminated end of the line to be reached, the line would behave just like a constant load, equal in value to the characteristic impedance of the line. However as the end of the line is reached, the electrons have nowhere to go hence they pile up at the end and then travel back to the battery at which point the current ceases and the line acts as a simple open circuit. This is shown in Figure 2.4-2) and described in more details in section 2.5.1. All this happens very quickly! For a 1km- long cable with a velocity factor of 0.66 (signal propagation velocity is 66% of light speed, or 200,000 km per second), it takes only 1/200,000 of a second (5μs) for a signal to travel from one end to the other. For the current signal to reach the line's end and "reflect" back to the source, the round-trip time is twice this figure, or 10 μs. A signal propagating from the source-end to the load-end of a transmission line is called an incident wave. The signal propagating from the load-end back to the source-end is called a reflected wave. High-speed measurement instruments are able to measure the time that it takes the signal to reach the end of the line and come back to the source, and may be used for the purpose of determining a cable's length. This technique may also be used for determining the presence and location of a break in one or both of the cable's conductors, since a current will "reflect" off the wire break just as it would off the end of an open-circuited cable. Instruments designed for such purposes are called time-domain reflectometers (TDRs). The basic principle is identical to that of sonar range finding: generating a sound pulse and measuring the time it takes for the echo to return. A similar phenomenon takes place if the end of a transmission line is short-circuited as shown in Figure 2.4-3 and explained in more detail in section 2.5.2. Now, if we have a finite stretch of line with a characteristic impedance of 50Ω, this will behave as a resistor to a constant source of DC voltage for the brief time it takes for the signal to reach the end of the line. If at this end we then connect a resistor equal to the characteristic impedance of the line, the source will continue to see a 50Ω load. As we mentioned earlier, there is no difference from the point of view of the battery between a resistor eternally dissipating energy and an infinite line absorbing energy, it still sees the same constant load! Reflections are therefore eliminated. In essence, a terminating resistor matching the characteristic impedance of the transmission line makes the line appear infinitely long from the perspective of the source, because a resistor has the ability to eternally dissipate energy in the same way as a transmission line of infinite length is able to eternally absorb energy. Conquer Radio Frequency 43 www.cadence.com/go/awr

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