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2.14 The dreaded maths 117 which shows more explicitly how the speed at which the vectors rotate is affected by both the frequency of the wave and its speed of propagation . Although does not feature directly in (2.14-2), if we rewrite (section 2.9) as And substitute (2.14-3) into (2.14-2) we get ( ) ( ) ( ) From (2.14-4) it is apparent that the speed at which the two vectors rotate is related to both the frequency of the wave, i.e. the rate of change of the amplitude of the field, and the speed of propagation, . Equation (2.14-2) makes use of vectors in the complex plane to represent the wave. We may however wish to represent the wave in the time domain, without resorting to such vectors. This may be achieved by taking the real part of (2.14-2) by means of Euler's formulae (section 1.5.3.1) and is shown by equation (2.14-5). { ( ) } ( ) ( ) The magnetic component of our electromagnetic wave may be expressed in a form which is similar to that used for the electric field in (2.14-1) ( ) [ ) ] Equation (2.14-6) is derived by using one of Maxwell's curl equations 24 . In (2.14-6), the and components are subtracted rather than added, therefore we may expect a phase difference between magnetic and electric fields. Note that in this case we cannot use voltage and current to calculate the impedance and hence electric and magnetic field magnitudes must be used instead | | | | The wave impedance is equal to √ ⁄ and in free space √ Ω You may be wondering why the exponential associated with the forward travelling field has a minus sign whereas the one associated the field travelling backwards has a positive exponent. There is a good explanation for this and if you are not prepared to take this leap of faith please do refer to reference [8]. 24 (2.14-3) (2.14-4) (2.14-5) (2.14-6) Conquer Radio Frequency 117 www.cadence.com/go/awr