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where we have used the fact that Is PFP = P (Is F) P 1 , which can be demonstrated by direct multiplication In relation to the previous analysis of this section it is useful to note 1 the following facts The quantity (sI F) represents a matrix of transfer functions Before any pole-zero cancelations, the denominator of each transfer function is the polynomial d(s) = |sI F| , which is the determinant of (sI F) The system poles are the solutions to the equation d(s) = |sI F| = 0 The eigenvalues of F are the solutions of |sI F| = 0 Therefore, before pole-zero cancelations, the eigenvalues of F and the poles of the transfer function from U(s) to Y(s) are identical Example 311 Consider the state space system x = y = 0 1 6 5 1 1 x+ x 0 1 u

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Problem 17-4 Suppose a circuit has an absolute-value impedance Z of 100 , with a resistance R = 80 What is the power factor Simply find the ratio PF = R/Z = 80/100 = 08 = 80% Note that it doesn t matter whether the reactance in this circuit is capacitive or inductive Problem 17-5 Suppose a circuit has an absolute-value impedance of 50 , purely resistive What is the power factor Here, R = Z = 50 Therefore, PF = R/Z = 50/50 = 1 = 100% Problem 17-6 Suppose a circuit has a resistance of 50 and a capacitive reactance of 30 in series What is the power factor Use the cosine method First, find the phase angle Remember the formula: = arctan (X/R), where X is the reactance and R is the resistance Therefore, = arctan ( 30/50) = arctan ( 060) = 31 The power factor is the cosine of this angle; PF = cos ( 31 ) = 086 = 86% Problem 17-7 Suppose a circuit has a resistance of 30 and an inductive reactance of 40 What is the power factor Use the R/Z method Find the absolute-value impedance: Z 2 = R 2 + X 2 = 30 2 + 40 2 = 900 + 1600 = 2500 Therefore, Z = 25001/2 = 50 , so PF = R/Z = 30/50 = 060 = 60% This problem can be represented vectorially by a 30:40:50 right triangle, as shown in Fig 17-8 How Much of the Power Is True The preceding formulas allow you to figure out, given the resistance, reactance, and VA power, how many watts are true or real power, and how many watts are imaginary or reactive power This is important in RF equipment, because some RF wattmeters display VA power rather than true power When there is reactance in a circuit or system, the wattage reading is therefore exaggerated.

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The eigenvalues of F are the roots of |sI F| = 0 where (sI F) = Therefore, |sI F| = s2 + 5s + 6 = (s + 3)(s + 2) which shows that the eigenvalues are -2 and -3 The transfer function from u to y is computed as Y (s) 1 = H (sI F) G U (s) = 1 1 s 6 1 s+5

0 1 0 1

Problem 17-7 (The vertical and horizontal scale increments differ; this is a common practice in graphs, often done for illustration convenience)

s+5 1 6 s 1 1 (s + 3)(s + 2) (s + 1) (s + 3)(s + 2)

Problem 17-8 Suppose a circuit has 50 of resistance and 30 of inductive reactance in series A wattmeter shows 100 W, representing the VA power What is the true power First, calculate the power factor Suppose you use the phase-angle method Then:

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which has a zero at s = 1 and poles at s = 2, 3 Example 312 Consider the state space system 0 1 0 0 0 1 v + 0 u v = 0 0 2 2 1 y = 0 1 0 v The eigenvalues are 0, 1 j, because s 1 0 s 1 (sI F) = 0 0 2 (s + 2) which yields the following equation for the eigenvalues |sI F| = s3 + 2s2 + 2s = s(s + 1 + 1j)(s + 1 1j) The transfer function from u to y is computed as follows: Y (s) U (s) = H (sI F)

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