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Angle equals radius times negative constant Figure 3-5 shows an example of what happens in polar coordinates when we set the radius equal to a negative constant multiple of the angle As in the previous case, we get a pair of spirals, but they re upside-down with respect to the case when the constant is positive To see

Notice that 1 varies nonlinearly with all three flows. A dimensionless plot of Eq. (9.3) $ppears in Fig. 9.1. Compare it with the numerical example given in Fig. 3.7. If total flow and final temperature are both to be controlled by manipulat ing WI and JV2, coupling will exist between the loops. The relative-

p /2

3p /2

At the beginning of the chapter, we listed seven di erent ways we can fail. We can really boil these down to just four. Projects fail when we: Leave key people out of the project Make the wrong product or solve the wrong problem Forget the 1:10:100 rule Do not nish what we start

constant multiple of the angle, we get a pair of spirals upside-down relative to those for a positive constant multiple of the angle Illustration for Problem 4

how this works, you can trace around with rotating rays as we did in Fig 3-4 Be careful with the signs and directions! Remember that negative angles go clockwise, and negative radii go in the opposite direction from the way the angle is defined

1.0 FIG 9.1. A plot of dimensionless temperature vs. dimensionless flow displays a typically nonlinear function. I= F 1 0.5 I-P

6

Following the six keys discussed in this chapter ensures that we do not make these four mistakes. Without these mistakes, we create a product or service everyone wants, we create it e ciently, and we deliver it successfully and ensure customer satisfaction. Let us take a look at the six keys once more by applying them to your project for learning.

.

Note that in the expression j 2ab, the numeral 2 is not an exponent! Now let s find the square of the complex number a jb Paying careful attention to the signs, we get (a jb)2 = (a jb)(a jb) = a2 + a( jb) + ( jb)a + ( jb)2 = a2 jab jba + ( j )2b2 = a2 j 2ab b2 = (a2 b2) j 2ab The two final products we ve derived are (a2 b2) + j 2ab and (a2 b2) j 2ab which, by definition, are complex conjugates 6 First, let s find the product of the polar complex vectors (p /4,21/2) and (3p /4,21/2) We must add the direction angles and multiply the magnitudes The sum of the angles is p /4 + 3p /4 = p The product of the magnitudes is 21/2 21/2 = 2 Therefore, the product vector is (p,2) The angle q is equal to p, and the magnitude r is equal to 2 To convert this polar vector (q,r) = (p,2) to the complex-number form a + jb where a and b are real-number coefficients, we use the formula for that purpose, getting a + jb = r cos q + j(r sin q) = 2 cos p + j(2 sin p) = 2 ( 1) + j 2 0 = 2 + j0 = 2 The product of the two original polar complex vectors (p /4,21/2) and (3p /4,21/2) is a vector representing the pure real number 2 7 Let s convert the polar vector (q,r) = (p /4,21/2) to a complex number in the traditional real-plus-imaginary form The conversion formula tells us that r cos q + j(r sin q) = 21/2 cos (p /4) + j[21/2 sin (p /4)] = 21/2 21/2/2 + j(21/2 21/2/2) = 1 + j Repeating the process with the polar vector (q,r) = (3p /4,21/2), we get r cos q + j(r sin q) = 21/2 cos (3p /4) + j[21/2 sin (3p /4)] = 21/2 ( 21/2/2) + j(21/2 21/2/2) = 1 + j.

Remember the project for learning you chose in 1 It is time to dust it o . Think about your project in terms of each of the six keys. On this project, think about how you will: Manage communication and teamwork De ne the project clearly Create a detailed work plan Ensure high quality Use written change control Follow through, deliver quality