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Forming the sum and di erence of eqns (899) and (8100) results in 1 + 2 = f1 f2 f1 f2 + + Ia + (N2 + N1 ) c c c c f2 f1 f1 f2 + + (8101) Ecm + 1 + 2 c c c c f1 f2 f1 f2 Ia + (N1 N2 ) c c c c f2 f1 f2 f1 + (8102) Ecm + 1 2 c c c c

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Pictorial Method It isn t necessary to construct an entire RL plane to find phase angles You can use a ruler that has centimeter (cm) and millimeter (mm) markings, and a protractor First, draw a line a little more than 10 cm long, going from left to right on a sheet of paper Use the ruler and a sharp pencil Then, with the protractor, construct a line off the left end of this first line, going vertically upward Make this line at least 10 cm long The horizontal line, or the one going to the right, is the R axis of a coordinate system The vertical line, or the one going upward, is the XL axis If you know the values of XL and R, divide them down or multiply them up so they re both between 0 and 100 For example, if XL = 680 and R = 840 , you can divide them both by 10 to get XL = 68 and R = 84 Plot these points lightly by making hash marks on the vertical and horizontal lines you ve drawn The R mark in this example will be 84 mm to the right of the origin, and the XL mark will be 68 mm up from the origin Next, draw a line connecting the two hash marks, as shown in Fig 13-12 This line will run at a slant, and will form a triangle along with the two axes Your hash marks, and the origin of the coordinate system, form the three vertices of a right triangle The triangle is called right because one of its angles is a right angle (90 ) Measure the angle between the slanted line and the R axis Extend one or both of the lines if necessary in order to get a good reading on the protractor This angle will be between 0 and 90 , and represents the phase angle in the RL circuit The complex impedance vector, R + jXL, is found by constructing a rectangle using the origin and your two hash marks as three of the four vertices, and drawing new horizontal and vertical lines to complete the figure The vector is the diagonal of this rectangle, as shown in Fig 13-13 The.

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1 2

method of finding phase angle in a circuit containing resistance and inductive reactance This method shows the actual impedance vector

By de ning the wide and narrow lane wavelengths as w n = = c f1 f2 c , f1 + f2 (8103) (8104)

eqns (8101) and (8102) can be written in meters as ( 1 + 2 ) n + Ecm Ia + (N2 + N1 ) n n n + 1 + 2 1 2 = + Ecm + Ia + (N1 N2 ) w w w + 1 2 1 2 =

phase angle is the angle between this vector and the R axis It will be the same as the angle of the slanted line in Fig 13-12

(8105)

(8106)

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Trigonometric Method If you have a good scientific calculator that can find the arctangent of a number (also called the inverse tangent and symbolized either as arctan or tan 1), you can determine the RL phase angle more precisely than the pictorial method allows Given the values of XL and R, the RL phase angle is the arctangent of their ratio Phase angle is symbolized by the lowercase Greek letter phi (pronounced fie or fee and written ) Therefore:

where w = 154 45 and w = 120 35; therefore, the standard devi 1 34 2 34 ation of the widelane noise (ie, w 1 w 2 ) is approximately 57 times 1 2 the standard deviation of the L1 or L2 phase noise (ie 1 or 2 ) The

= tan 1 (XL/R) or = arctan (XL/R)

f (2 f )

Based on the de ning constants in Table 21, the following ellipse parameter values can be derived: f b e = 000335281, = a(1 f ) = 6356752314m, = f (2 f ) = 008181919 (23) (24) (25)

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