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1.5 The significance of reactive components equations 21 1.5.4.2 Digression on Euler's Formulae As we have seen in previous sections, Euler's formulae (1.5-11) are extremely useful to express mathematically what happens in circuits where periodic stimuli are present. In this section I would just like to give the reader a bit more insight into these formulae and show that an appropriate graphical representation greatly helps with understanding where they come from. Let us consider equation (1.5-11)(a), Now, you will recall, from the physics of rotational motion, that if we are moving around a circle at an angular speed , the angle which we travel in time is equal to . Equation (1.5-11)(a) may therefore be rewritten as The term may be represented in the complex plane, as a vector rotating anticlockwise as time increases, with angular speed . Equally the term represents a vector in the complex plane, rotating clockwise at radians per second. This is illustrated in Figure 1.5-9. So the numerator of equation (1.5-13) is just the sum of these two vectors of equal magnitudes which are rotating at the same speed but in opposite directions. From Figure 1.5-9 (a)-(f), it is apparent that the sum of these two vectors always lies on the real axis. If we plot the modulus of this sum on the y-axis of an ordinary Cartesian graph versus time, as shown on the right-hand pane of Figure 1.5-9 (a)-(f), then we can see that we get a cosine-like function. This is also illustrated in an animated fashion in video 1.5 You may notice however that this function varies between -2 and 2, i.e. is twice the amplitude of a cosine function. This is why equation (1.5-13) prescribes that we should halve it to get the cosine. Understanding complex exponentials and what they represent is key to understanding the mathematical representation of waves and their propagation though transmission lines and media. As we will see in section 2.14, just as light shining through a glass is partly reflected and partly shone through, electromagnetic waves and signals undergo the same process at radio frequency. The reflected and transmitted signals, which travel in opposite directions, may be represented by complex exponentials rotating around in opposite directions on the complex plane, in a similar fashion to the ones shown in Figure 1.5-9. For sine functions we may use a similar equation And an analogous argument applies as shown in this animation (video 1.6) (1.5-11)(a) (1.5-13) (1.5-11)(b) Conquer Radio Frequency 21 www.cadence.com/go/awr