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Figure 211: De nition of i for i = 1, 2, 3 in eqn (216) Because each element of R1 is the cosine of the angle between one of 2 the coordinate axes of 1 and one of the coordinate axes of 2 , the matrix R1 is referred to as a direction cosine matrix: 2 cos( 1 ) cos( 1 ) cos( 1 ) R1 = cos( 2 ) cos( 2 ) cos( 2 ) (216) 2 cos( 3 ) cos( 3 ) cos( 3 ) Figure 211 depicts the angles i for i = 1, 2, 3 that de ne the rst column of R1 The i and i angles are de ned similarly When the relative 2 orientation of two reference frames is known, the direction cosine matrix R1 is unique and known 2 Although the direction cosine matrix has nine elements, due to the three orthogonality constraints and the three normality constraints, there are only three degrees of freedom Alternative parameterizations are discussed in Section 25 and Appendix D Analogous to eqn (213) v2 = I2 x2 + J2 y2 + K2 z2 where [v2 ]2 = [x2 , y2 , z2 ] and by eqn (214) I1 1 [v2 ] = J1 v2 K1

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Therefore, [v2 ]

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Step-down Most solid-state electronic devices, such as radios, need only a few volts The power supplies for such equipment use step-down power transformers The physical size of the transformer depends on the current Some devices need only a small current and a low voltage The transformer in a radio receiver, for example, can be physically small A ham radio transmitter or hi-fi amplifier needs more current This means that the secondary winding of the transformer must consist of heavy-gauge wire, and the core must be bulky to contain the magnetic flux Step-up Some circuits need high voltage The cathode-ray tube (CRT) in a conventional home television set needs several hundred volts Some ham radio power amplifiers use vacuum tubes working at more than 1 kV dc The transformers in these appliances are step-up types They are moderate to large in size, because of the number of turns in the secondary, and also because high voltages can spark, or arc, between wire turns if the windings are too tight If a step-up transformer needs to supply only a small amount of current, it need not be big But for ham radio transmitters and radio or television broadcast amplifiers, the transformers are large, heavy, and expensive Transformer Ratings Transformers are rated according to output voltage and current For a given unit, the volt-ampere (VA) capacity is often specified This is the product of the voltage and current

I1 = J1 (I2 x2 + J2 y2 + K2 z2 ) K1 I1 I2 I1 J2 I1 K2 = J1 I2 x2 + J1 J2 y2 + J1 K2 z2 K1 I2 K 1 J2 K1 K2

A transformer with 12-V output, capable of delivering 10 A, has 12 V 10 A = 120 VA of capacity The nature of power-supply filtering, to be discussed later in this chapter, makes it necessary for the power-transformer VA rating to be greater than the wattage consumed by the load A high-quality, rugged power transformer, capable of providing the necessary currents and/or voltages, is crucial in any power supply The transformer is usually the most expensive component to replace

24 REFERENCE FRAME TRANSFORMATIONS = R1 [v2 ] 2

39 (217)

Rectifier diodes are available in various sizes, intended for different purposes Most rectifier diodes are made of silicon, and are known as silicon rectifiers Some are fabricated from selenium, and are called selenium rectifiers Two important features of a power-supply diode are the average forward current (Io) rating and the peak inverse voltage (PIV) rating

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