Teach Yourself Electricity and Electronics, 5th edition

Stan Gibilisco

Explanations for Quiz Answers in Chapter 14

1. If we have a theoretically ideal circuit containing no resistance and a finite, nonzero capacitive reactance, then the current leads the voltage by 90º. Because the reactance is capacitive rather than inductive, we say, by convention, that the phase angle is negative, so it equals -90º in this case. The correct answer is (d).

2. When we have a circuit in which the resistance and the absolute value of the capacitive reactance are equal and nonzero, the current leads the voltage by 45º. That means the phase angle equals -45º. We can figure this out from the formula for the phase angle in an RC circuit:

f = Arctan (X_C / R)

where f represents the phase angle in degrees, X_C represents the capacitive reactance, and R represents the resistance (in the same units as the capacitive reactance). We know that in this particular case, the ratio X_C / R equals -1:1, so

f = Arctan (-1/1)
= Arctan -1
= -45º

The correct answer is (c).

3. In a circuit containing pure resistance with no capacitive reactance, the current and the voltage coincide in phase (assuming that the circuit has no inductive reactance). Therefore, the phase angle equals 0º. The correct choice is (b).

4. Let's remember the formula for capacitive reactance 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 reactance X_C (in ohms) using the approximate formula

X_C = -1 / (6.2832 fC)

This formula also works for values of f in megahertz and values of C in microfarads. We've been told that C = 500 pF and f = 2.55 MHz. We should convert the capacitance to microfarads for consistency of units. That gives us C = 0.000500 µF. When we plug our values into the above formula, we get

X_C = -1 / (6.2832 x 2.55 x 0.000500)
= -125 ohms

The correct choice is (a).

5. We can use the formula from the solution to Question 4 to solve this problem, but we'll have to do a little algebra to get the answer. We're told that C = 0.0100 µF and X_C = -75.0 ohms. If we plug these values directly into the formula

X_C = -1 / (6.2832 fC)

and then solve for f, we'll get a result in megahertz. Let's do that, and then convert to kilohertz when we're done with the calculations! Plugging in the numbers yields

-75.0 = -1 / (6.2832 x f x 0.0100)

When we negate both sides of this equation, we get

75.0 = 1 / (6.2832 x f x 0.0100)

Multiplying through by f gives us

75.0 f = 1 / (6.2832 x 0.0100)
= 15.915

Dividing through by 75.0, we get the answer as f = 0.212 MHz. That's 212 kHz, so the correct answer is (c).

6. As the ratio X_C / R increases negatively without limit, the phase angle, which equals the Arctangent of the ratio X_C / R, approaches -90º. The correct choice is (a).

7. If we increase the spacing between the plates of a capacitor while leaving all other factors unchanged, the capacitance decreases. From the formula for capacitive reactance X_C in terms of the frequency f and the capacitance C, assuming a constant value of f and a diminishing value of C, we can see that X_C increases negatively. The correct answer is therefore (a).

8. Let's use the formula for capacitive reactance, inputting the value of f in megahertz and the value of C in microfarads:

X_C = -1 / (6.2832 fC)

We have to convert both the capacitance and the frequency to the proper units in order for this formula to work here. We know that C = 100 pF and f = 159.15 kHz. The capacitance value equals 0.000100 µF. The frequency equals 0.15915 MHz. When we plug these numbers into our formula, we get

X_C = -1 / (6.2832 x 0.15915 x 0.000100)
= -10,000 ohms

When we convert this result to kilohms and then round it off to three significant figures (the extent of the input accuracy), we get X_C = -10.0 k. The correct choice is (d).

9. If we double the frequency, we double the denominator in the formula. Therefore, the capacitive reactance drops to 1/2 its previous value (negatively, of course). The correct answer is (b).

10. We can use the formula for capacitive reactance here. However, as in the solution to Question 5, we must work through some algebra. We're told that X_C = -433 ohms and f = 433 kHz. If we convert to megahertz, we get f = 0.433 MHz, so our answer will come out in microfarads. (We can convert to picofarads, the units that appear in all the answer choices, after we've finished our calculations.) Let's plug X_C = -433 and f = 0.433 into the formula

X_C = -1 / (6.2832 fC)

and then solve for C. We start with

-433 = -1 / (6.2832 x 0.433 x C)

When we negate both sides, we get

433 = 1 / (6.2832 x 0.433 x C)

Multiplying through by C gives us

433 C = 1 / (6.2832 x 0.433)
= 0.36756

Dividing through by 433, we obtain C = 0.000849 µF. That's the same as 849 pF, so the correct answer is (c).

11. To solve this problem, we must use the formula for phase angle in terms of capacitive reactance X_C and resistance R. Once again, that formula is

f = Arctan (X_C / R)

where f represents the phase angle in degrees. We have X_C = -75 ohms and R = 100 ohms. Therefore

f = Arctan (-75/100)
= Arctan -0.75
= -37º

The correct choice is (b).

12. If we double the capacitive reactance (negatively) while leaving all other factors constant in the above-described scenario, we cause the ratio X_C / R to increase (negatively) to -150/100, or -1.5. The Arctangent of -1.5 equals approximately -56º, which is larger negatively than the previous angle. The correct choice is therefore (a).

13. We must solve this problem in two steps. First, let's find the capacitive reactance, inputting the value of f in megahertz and the value of C in microfarads:

X_C = -1 / (6.2832 fC)

When we convert the frequency of 770 kHz to megahertz, we get f = 0.770 MHz. We have been told that C = 0.047 µF. When we plug these numbers in, we get

X_C = -1 / (6.2832 x 0.770 x 0.047)
= -4.3977 ohms

which rounds off to -4.40 ohms. We know that the resistance has a value of R = 4.40 ohms. We can now carry out the second step, calculating the phase angle as

f = Arctan (X_C / R)
= Arctan (-4.40/4.40)
= Arctan -1
= -45º

The correct choice is (c).

14. If we double the resistance while leaving all other factors constant in the scenario of Question 13, we cut the ratio X_C / R in half (negatively). That causes the Arctangent to decrease negatively (get closer to 0º). The correct choice is (d).

15. If we short out the resistor in the situation of Question 13, we're left with only the capacitance. Therefore, the current will lead the voltage by 90º for all practical purposes. The phase angle will equal essentially -90º. The correct answer is (a).

16. If we double the frequency and change nothing else in the situation of Question 13, we cut the capacitive reactance in half (negatively), so we cut the ratio X_C / R in half (negatively). That causes the Arctangent to decrease negatively (get closer to 0º). The correct answer is (d).

17. We're told that C = 680 pF and X_C = -680 ohms. The capacitance is equivalent to 0.000680 µF. If we plug these values directly into the formula

X_C = -1 / (6.2832 fC)

we can solve for f in megahertz, which we can convert to kilohertz at the end. Plugging in the numbers yields

-680 = -1 / (6.2832 x f x 0.000680)

When we negate both sides of this equation, we get

680 = 1 / (6.2832 x f x 0.000680)

Multiplying through by f gives us

680 f = 1 / (6.2832 x 0.000680)
= 234

Dividing through by 680, we get the answer as f = 0.344 MHz. That's 344 kHz, so the correct answer is (a).

18. We're told that X_C = -870 ohms and f = 4.78 MHz. We can plug these values directly into the formula

X_C = -1 / (6.2832 fC)

and then solve for C in microfarads. We start with

-870 = -1 / (6.2832 x 4.78 x C)

When we negate both sides, we get

870 = 1 / (6.2832 x 4.78 x C)

Multiplying through by C gives us

870 C = 1 / (6.2832 x 4.78)
= 0.033296

Dividing through by 870, we obtain C = 0.0000383 µF. That's equivalent to 38.3 pF, so the correct choice is (c).

19. We can see from Fig. 14-13 that the capacitive reactance equals approximately -82 ohms and the resistance equals approximately 29 ohms. Therefore, the ratio X_C / R equals approximately -82/29, or -2.83. The correct answer is (a).

20. We can find the phase angle by taking the Arctangent of X_C / R. In this case, that ratio equals approximately -2.83. Using a calculator to find the Arctangent of -2.83, we get approximately -71º. The correct choice is (d).