Teach Yourself Electricity and Electronics, 5th edition |

Stan Gibilisco |

Explanations for Quiz Answers in Chapter 15 |

1. When we take the positive square root of a negative real number, we get a positive
imaginary number. That always turns out as a positive real-number multiple of j, so
the correct answer is (c). |

2. The reciprocal of j equals its negative. That is,1 / The correct choice is (b). |

3. If we add a real number to an imaginary number, we get a complex number. The correct choice is (c). |

4. When we want to add two complex numbers, we add their real parts and their
imaginary parts separately and then put the results back together to form a new complex
number. In this case, we want to find (-1 + j7) + (3 - j5). The real parts
are -1 and 3; when we add them we get -1 + 3 = 2. The imaginary partss are j7 and -j5; when
we add them we get j7 + (-j5) = j2. Therefore, the sum we seek equals 2 + j2. The
correct answer is (a). |

5. We want to find (3 - j5) + (-1 + j7). The real parts are 3 and -1;
when we add them we get 3 + (-1) = 2. The imaginary parts are -j5 and j7; when we add them
we get -j5 + j7 = j2. Therefore, the sum we seek equals 2 + j2. The correct answer is
(a). In case you're curious, complex-number addition obeys the commutative law,
meaning that we can add two complex numbers in either order, and get the same result. |

6. When we want to subtract one complex number from another, we subtract their real
parts and their imaginary parts separately and then put the results back together, making
sure we perform the subtractions in the correct order. In this case, we want to find (-1 +
j7) - (3 - j5). The real parts are -1 and 3; when we subtract them in this
order we get -1 - 3 = -4. The imaginary parts are j7 and -j5; when we subtract them in this
order we get j7 - (-j5) = j7 + j5 = j12. The difference we seek equals -4 + j12, so the
correct choice is (c). |

7. We want to find (3 - j5) - (-1 + j7). The real parts are 3 and -1;
when we subtract them in this order we get 3 - (-1) = 3 + 1 = 4. The imaginary parts are
-j5 and j7; when we subtract them in this order we get -j5 - j7 = -j12. The difference we seek
equals 4 - j12, so the correct answer is (b). Again, if you're curious and astute,
you might want to "memorize" the fact that complex-number subtraction obeys the anti-commutative
law. That's a mathematician's way of saying that if we reverse the order of a
difference between two complex numbers, we reverse the signs of both the real and the
imaginary components of the result. |

8. Whenever you read or hear that a certain impedance equals a fixed real number of
ohms (such as Z = 50), you can reasonably assume that the author or manufacturer
means a pure resistance of that number of ohms. None of the three choices (a), (b), or (c)
meet that criterion, so we must choose (d), "None of the above." |

9. The complex impedance 15 + j15 indicates 15 ohms of resistance and 15 ohms
of inductive reactance. A resistor in series with an inductor could exhibit that property,
so the correct choice is (c). |

10. In order to determine the absolute value of a complex number, we must square the
real part, then square the real-number coefficient of the imaginary part (that's the
number after the j taking the sign into account), then add the two squares, and
finally take the square root of the sum of the squares. If we call the absolute value Z
and then carry out the foregoing arithmetic exercise with choice (a), we get
That's the value we seek! We don't get that result for the complex numbers given at either (b) or (c), so we can conclude that the correct choice is (a). |

11. When we square the real-number part (which equals 4.50) of this complex impedance, we get 20.25. When we square the real-number coefficient of the imaginary part (which equals 5.50), we get 30.25. Adding the two squares gives us 50.50. When we take the square root of that quantity and then round off to three significant figures, we obtain 7.11 ohms. The correct choice is (c). |

12. When we square the real-number part (which equals 0.0), we get 0.0. When we square the real-number coefficient of the imaginary part (which equals -36), we get 1296. Adding the two squares gives us 1296. When we take the square root, we get 36 ohms. The correct choice is (d). |

13. When we square the real-number part (which equals 1000), we get 1,000,000. When we square the real-number coefficient of the imaginary part (which equals -1000), we get 1,000,000. Adding the two squares gives us 2,000,000. When we take the square root and round off to four significant figures, we get 1414. The correct choice is (b). |

14. Remember that complex numbers in mathematics can have negative real-number parts, even though complex impedances in electricity and electronics normally can't. When we square the real-number part (which equals -1000), we get 1,000,000. When we square the real-number coefficient of the imaginary part (which equals -1000), we get 1,000,000. Adding the two squares gives us 2,000,000. When we take the square root and round off to four significant figures, we get 1414. The correct choice is (b). This situation represents a "mirror image" of the scenario described in Question 13. |

15. If we increase the inside radius of the cylindrical shield in a coaxial cable while leaving all other factors the same, the characteristic impedance gets larger. The correct choice is (a). |

16. If we increase the radii of both wires in a parallel-wire transmission line while leaving all other parameters the same, the characteristic impedance goes down. The right choice is (d). |

17. Let's remember the formula for capacitive susceptance in terms of frequency and
capacitance. If the frequency of an AC source equals f (in hertz) and the
capacitance of a component equals C (in farads), then we can calculate the
capacitive susceptance B (in siemens) using the approximate formula_{C}
fCThis formula also works for values of
= 0.075 S In imaginary terms, that's |

18. When we want to find the absolute-value impedance based on a complex expression in terms of resistance and reactance, we square the resistance, square the real-number coefficient of the reactance, add the two squares, and finally take the square root of the result. None of the choices (a), (b), or (c) says any such thing, so the correct answer is (d), "None of the above." |

19. Let's remember the formula for inductive susceptance in terms of frequency and
inductance. If the frequency of an AC source equals f (in hertz) and the inductance
of a component equals L (in henrys), then we can calculate the inductive
susceptance B (in siemens) using the approximate formula_{L}
fL)This formula also works for values of
= -0.00100 S In imaginary terms, that's - |

20. The reciprocal of real-number resistance (in ohms) equals real-number conductance (in siemens). The reciprocal of imaginary-number reactance (in ohms) equals imaginary-number susceptance (in siemens). When we add real-number conductance to imaginary-number susceptance, we get complex-number admittance. The correct answer is (d). |