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CHAPTER 2 - Conveying Power at Radio Frequency 116 2.14 The dreaded maths 2.14.1 Introduction The ability of electromagnetic waves to propagate comes from two fundamental laws of physics which are part of Maxwell's equations: - Faraday's law, which states that a changing magnetic field causes an electric field, and - Ampere's law, which states that a changing electric field causes a magnetic field Once these fields are created, they sustain one another and propagate through space at the speed of light (in free space). This concept was introduced in section 2.1. What we will be looking at here, are the mathematical models which are used to represent travelling waves both in free space and in transmission lines. This section by no means aims to present a comprehensive treatment of this topic which could fill an entire book! But we would like to give the reader an idea of the maths behind the concepts explained in previous sections and show how it all ties in. We will only be considering plane waves propagating in TEM mode (section 2.11.2) i.e. waves in which electric and magnetic fields are in the same xy plane and perpendicular to one another and also perpendicular to the direction of propagation z. This is shown in Figure 2.14-1. Figure 2.14-1 Plane Wave As show in Figure 2.14-1, the electric field is a vector, varying in magnitude with time and whose direction is confined to the y-axis. The electric field, for a wave of frequency , may be represented mathematically by equation (2.14-1). ( ) [ ] Now, as you may recall from section 1.5.4.2, and represent two vectors on the complex plane, rotating around at the same speed but in opposite directions. Also the envelope of the resultant sum of these two exponentials has a sinusoidal behaviour (section 1.5.4.2). In this case however, and are also multiplied by different constants, and , which means that the amplitude of these rotating vectors may not be the same. In addition, they are both multiplied by the same exponential which is used to express the dependency of the field on the frequency of the propagating signal. We may rewrite (2.14-1) as ( ) ( ) ( ) x y z (2.14-1) (2.14-2) Conquer Radio Frequency 116 www.cadence.com/go/awr