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2.9 Transmission Lines Applied to High Frequency Circuits 93 2.9 Transmission Lines Applied to High Frequency Circuits In section 2.8.3 we saw how, when a line is terminated with its characteristic impedance, no reflections occur and the impedance seen at the generator end is the same as the characteristic impedance of the line, irrespective of its length. When there is a mismatch, however, as we have seen in sections 2.8.1, 2.8.2, 2.8.4 and 2.8.5, reflections do occur and the amplitude of the total voltage (incident plus reflected) changes all the time. Its envelope however remains constant and a standing wave may be observed whose peaks and nulls remain 'anchored' to fixed positions along the line. Imagine that we were able to probe the mismatched transmission line at any point along its length and observe voltage and current waveforms at each point. This would then enable us to work out the impedance at that point. It would be difficult to do this in practice but we could do it with a simulator (video 3.3). The great thing about simulation is that it allows us to see exactly what is happening at any position and hence gain some great insight into the behaviour of the line. We will do this in due course but let us first start with the mathematical representation of a transmission line, terminated with any impedance, as seen from its input terminal (Figure 2.9-1). Figure 2.9-1 Input Impedance of a Transmission line of characteristic impedance Z 0 terminated with load impedance Z L For a lossless transmission line may be represented by equation (2.9-1). ( ) ( ) Where is the length of the line and is the propagation constant. , which is equal to ⁄ , is used to account for the specific speed of propagation at which the signal travels in the transmission line of interest. Note that, although the speed of propagation does not appear explicitely in the expression for , at a specific frequency, and may interchangebly be used (eq (2.2-4)). We will be looking at how equation (2.9-1) is derived in section 3.5 but for now let us just use it to understand the behaviour of our transmission line. First of all let us take a look at the ( ) function shown in Figure 2.9-2. Notice that the periodicity of the trigonometric tangent is just and not 2 as in sine and cosine functions. This periodicity is one of the features of this trigonometric function which makes it suitable to represent the impedance of transmission lines, as we will in section 2.10. Z L V S Z 0 Transmission Line R S Z IN (2.9-1) Conquer Radio Frequency 93 www.cadence.com/go/awr

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