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Module 2 : Transmission Lines Lecture 13 : Application of Transmission Lines continues Objectives In this course you will learn the following What is the resonant section of a transmission line? Frequency response of a resonant section of a line. Input impedance of a resonant section of a line. Voltage and current on a resonant section of a line. Module 2 : Transmission Lines Lecture 13 : Application of Transmission Lines continues Transmision Lines as Resonant Circuits If the length of a short or open circuited line is exact multiple of , the imput impedance of the line is zero or us plot the input impedance as a function of frequency ` ', for a given length of transmission (short circuit or open circuit). . Let and a given termination Figure shows the variation of reactance as a function of frequency for open and short circuited sections of a transmission line. It is clear that around frequencies .., for which the length is an integer multilple of , the impedance variation is identical to an L-C resonant circuit. In the vicinity of these frequencies the line can be used as a LC - resonant circuit. Module 2 : Transmission Lines Lecture 13 : Application of Transmission Lines continues Frequency response of Resonant Circuit The impedance characteristics of a series and a parallel resonant circuit are shown in the figure below Comparing Figure (1) with Figure (2), one can observe that a short circuited line behaves like a parallel resonant circuit around frequencies and , whereas around and its behaviour is like a series resonant circuit. In general a short circuited section of a line is equivalent to a parallel resonant circuit. Similarly, the line is equivalent to a series resonant circuit. A converse is true for an open circuited section of a line i.e., if the length of the line is equal to odd multiples of line behaves like a series resonant circuit, and if the length of the line is equal to even multiple of like a parallel resonant circuit. , the , the line behaves Module 2 : Transmission Lines Lecture 13 : Application of Transmission Lines continues Input Impedance of Resonant Line Input impedance of a resonant lossless line is either . However, in practice, the lines have finite loss. This loss has to be included in the calculations while analysing the resonant lines. The complex propagation constant in impedance calculations of a resonant line. The input impedance of a short or open circuited line having propagation constant Note that although has to be used can be written as has been taken complex for a low-loss transmission line, is almost real. Substituting for , we get For a low-loss line, taking , we have . Also . Hence we get Similarly for an open circuited line we get For resonant lines, , is integer multiples of i.e., is integer multiples of . If we take odd multiples of , and we get On the other hand if we take even multiples of , , giving Conclusion A parallel resonant section of a line has an impedance and a series resonant section has an impedance . One can cross-check the result with that of an ideal loss-less line. In the absence of any loss the parallel resonant circuit shows infinite impedance and a series resonant circuit shows zero impedance at the resonance. Module 2 : Transmission Lines Lecture 13 : Application of Transmission Lines continues Voltage & Current on a Resonant Section of a line Consider a short circuited section of a line having length equal to odd multiples of resonant circuit. Let the line be applied with a voltage . This line is equivalent to a parallel between its input terminals as shown in Figure(a). The voltage and current standing wave patterns on the line are shown in Figure(b,c). The voltage is zero at the short-circuit-end of the line and is maximum at the input end of the line. similarly, the current is maximum at the short-circuit end and minimum at the input end of the line. The maximum value of the voltage on the line is the voltage and current on the line are given as and maximum value of current is . For a short-circuited line Module 2 : Transmission Lines Lecture 13 : Application of Transmission Lines continues Recap In this course you havel learnt the following What is the resonant section of a transmission line? Frequency response of a resonant section of a line. Input impedance of a resonant section of a line. Voltage and current on a resonant section of a line. Module 2 : Transmission Lines Lecture 13 : Application of Transmission Lines continues Recap In this course you havel learnt the following What is the resonant section of a transmission line? Frequency response of a resonant section of a line. Input impedance of a resonant section of a line. Voltage and current on a resonant section of a line.