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You can use a protractor and a ruler to find phase angles for RC circuits, just as you did with RL circuits, as long as the angles aren t too close to 0 or 90 degrees. First, draw a line somewhat longer than 10 cm, going from left to right on the paper. Then, use the protractor to construct a line going somewhat more than 10 cm vertically downwards, starting at the left end of the horizontal line. The horizontal line is the R axis of a crude RC plane. The line going down is the XC axis. If you know the values of XC and R, divide or multiply them by a constant, chosen to make both values fall between 100 and 100. For example, if XC 3800 and R 7400 , divide them both by 100, getting 38 and 74. Plot these points on the lines as hash marks. The XC mark goes 38 mm down from the intersection point between your two axes (negative 38 mm up from the intersection point). The R mark goes 74 mm to the right of the intersection point. Now, draw a line connecting the two hash marks, as shown in Fig. 14-11. This line will be at a slant, and will form a triangle along with the two axes. This is a right triangle, with the right angle at the origin of the RC plane.

When you add a real number and an imaginary number, you get a complex number. In this context, the term complex does not mean complicated. A better word would be composite. Examples are

DS1/DS2/DS3 E1/E3 25.6-Mbps User-to-Network Interface (UNI) over UTP-3 51-Mbps UNI over UTP-5 (Transparent Asynchronous Transmitter/Receiver Interface [TAXI]) 100-Mbps UNI over UTP-5 OC3/12/48c Others, of course, will follow as transport technologies advance.

Measure the angle between the slanted line and the R, or horizontal, axis. Use the protractor for this. Extend the lines, if necessary, using the ruler, to get a good reading on the protractor. This angle will be between 0 and 90 degrees. Multiply this reading by

4 + j5, 8 j 7, 7 + j13, and 6 j 87. The set of complex numbers needs two dimensions a plane to be graphically defined.

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258 Capacitive reactance 1 to get the RC phase angle. That is, if the protractor shows 27 degrees, the RC phase angle is 27 degrees. The actual vector is found by constructing a rectangle using the origin and your two hash marks, making new perpendicular lines to complete the figure. The vector is the diagonal of this rectangle, running out from the origin (Fig. 14-12). The phase angle is the angle between the R axis and this vector, multiplied by 1. It will have the same measure as the angle of the slanted line you constructed in Fig. 14-11.

As mentioned before, ATM is a cell-based technology that relies on a 48octet payload field that contains actual user data, and a five-byte header that contains information needed by the network to route the cell and provide proper levels of service. The ATM cell header, shown in Figure 6-7, is examined and updated by each switch it passes through and comprises six distinct fields: the Generic Flow Control (GFC) field, the Virtual Path Identifier (VPI), the Virtual Channel Identifier (VCI), the Payload Type Identifier (PTI), the Cell Loss Priority (CLP) field, and the Header Error Control (HEC) field.

The more accurate way to find RC phase angles is to use trigonometry. Then you don t have to draw a figure and use a protractor. You only need to punch a few buttons on your calculator. (Just make sure they re the right ones, in the right order.) If you know the values XC and R, find the ratio XC /R. If you know the ratio only, call it XC /R and enter it into the calculator. Be sure not to make the mistake of getting the ratio upside down (R/XC). This ratio will be a negative number or zero, because XC is always negative or zero, and R is always positive. Find the arctangent (arctan or tan 1) of this number. This is the RC phase angle. Always remember, when doing problems of this kind, to use the capacitive reactance for XC, and not the capacitance. This means that, if you are given the capacitance, you must use the formula for XC and then calculate the RC phase angle.

Adding and Subtracting Complex Numbers Adding complex numbers is just a matter of adding the real parts and the complex parts separately. For example, the sum of 4 + j 7 and 45 j 83 works out like this:

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