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In some instances, a matrix will be de ned with respect to a speci c frame of reference Eqns (221 222) can be used to derive the transformation of such matrices between frames-of-reference a a Let a R3 3 be a matrix de ned with respect to frame a and v1 , v2 3 R be two vectors de ned in frame a Let b be a second frame of reference a a If v1 and v2 are related by

a a v1 = a v2 ,

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Transconductance Recall the discussion of dynamic current amplification from the last chapter This is a measure of how well a bipolar transistor amplifies a signal The JFET equivalent of this is called dynamic mutual conductance or transconductance Refer again to Fig 23-5 Suppose that EG is a certain value, resulting in a certain current ID If the gate voltage changes by a small amount d EG, then the drain current will change by a certain increment dID The transconductance is the ratio dID/dEG Geometrically, this translates to the slope of a line tangent to the curve of Fig 23-5 The value of dID/dEG is not the same at every point along the curve When the JFET is biased beyond pinchoff, in the region marked Y, the slope of the curve is zero Then there is no fluctuation in ID when EG changes by small amounts There can be a change in ID when there is a change in EG only when the channel conducts current The region where the transconductance, dID/dEG, is the greatest is the region marked X, where the slope of the curve is steepest This region of the curve represents conditions where the most gain can be obtained from the device

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where b = Rb a Ra a b is the representation of the matrix with respect to frame b (223)

The acronym MOSFET (pronounced moss-fet ) stands for metal-oxide-semiconductor field effect transistor A simplified cross-sectional drawing of an N-channel MOSFET, along with the schematic symbol, is shown in Fig 23-7 The P-channel device is shown in the drawings of Fig 23-8

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In Section 24, the transformation of vectors from frame a to frame b is shown to involve an orthonormal matrix denoted by Rb The elements of a this matrix, called the direction cosine matrix, are the cosines of the angles between the coordinate axes of the two frames-of-reference Although this appears to allow nine independent variables to de ne Rb , orthonormality a restrictions result in only three independent quantities Section 251 introduces the concept of a plane rotation Sections 252 255 will use plane rotations to de ne the transformations between speci c pairs of reference systems In addition to the direction cosine and Euler angle representations of the relative orientations of two reference frames, various other representations exist [120] Advantages of alternative representations may include e cient computation, lack of singularities, or compact representation One popular representation of relative attitude is the quaternion Quaternions o er accurate and e cient computation methods without singularities Often quaternions are preferred over both direction cosine and Euler angle methods Nonetheless, their discussion is a topic in and of itself To maintain the ow of the book, the discussion of quaternions has been placed in Appendix D It is recommended that designers read the main body of the text rst, to understand the role and issues related to attitude representation; however, they should understand and consider quaternions prior to implementation of their rst system

A plane rotation is a convenient means for mathematically expressing the rotational transformation of vectors between two coordinate systems where the second coordinate system is related to the rst by a rotation of the rst coordinate system by an angle x around a vector v In the special case where the vector v is one of the original coordinate axes, the plane rotation matrix takes on an especially simple form In the following, a rotation of the rst coordinate system by x radians1 around the i-th axis will be expressed as [x]i Using this notation, 1 0 0 cos(x) sin(x) [x]1 = 0 (224) 0 sin(x) cos(x) cos(x) 0 sin(x) 0 1 0 [x]2 = (225) sin(x) 0 cos(x)

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