Electronics Demystified, 2nd edition
Stan Gibilisco
Explanations for Quiz Answers in Chapter 3
1. When we want to add two complex numbers, we should add the real parts to each other, add the imaginary parts to each other, and then combine the results as a sum. In this case, we have

(5 + j7) + (-3 - j2)

The real parts add to give us

5 + (-3) = 2

and the imaginary parts add to produce

j7 + (-j2) = j5

The sum of the original two complex numbers is therefore

(5 + j7) + (-3 - j2) = 2 + j5

The correct choice is C.

2. To find the reciprocal of an imaginary number, we reverse the sign of the j operator and take the reciprocal of the real coefficient. If we start with j2, as in this case, we end up with -j(1/2), which equals -j0.5. The correct choice is B.
3. Let's start by calculating the reactance of the 100-H inductor at 318.31 kHz. We can change the frequency to 0.31831 MHz to go with microhenrys in the inductive-reactance formula. That formula is

XL = 6.28318 f L

where XL represents the inductive reactance in ohms, 6.28318 approximates twice the value of pi (the ratio of a geometric circle's circumference to its diameter), f represents the frequency in megahertz, and L represents the inductance in microhenrys. In this case, f = 0.31831 MHz and L = 100 H, so

XL = 6.28318 x 0.31831 x 100
= 200 ohms

The resistance R doesn't depend on the frequency; it's 100 ohms no matter what. We can now combine the resistance (as a real number) and the reactance (as an imaginary number) to get

R + jXL = 100 + j200

The correct choice is A.

4. First, let's find the reactance of a 1000-pF capacitor. We're told that the frequency is 3.1831 MHz. We can express the capacitance as 0.001000 F to go with megahertz in the formula

XC = -1 / (6.28318 f C)

where XC represents the capacitive reactance in ohms, f represents the frequency in megahertz, and C represents the capacitance in microfarads. In this case, f = 3.1831 MHz and C = 0.001000 F, so

XC = -1 / (6.28318 x 3.1831 x 0.001000)
= -50 ohms

The resistance R is 100 ohms, as before. We combine the resistance (as a real number) and the reactance (as an imaginary number) to get

R + jXC = 100 - j50

The correct choice is A.

5. Whenever we find resistance and reactance together in an AC circuit, we must know the frequency before we can determine the reactance. Because complex impedance contains components of resistance and reactance, we can't calculate the complex impedance unless we know the frequency. We aren't given any value for the frequency, so the correct choice is D, "We need more information to say."
6. The resistance has no effect on the resonant frequency of this circuit. Only the inductance L and the capacitance C have significance. If we express the inductance in microhenrys, we have L = 100. If we express the capacitance in microfarads, we have C = 0.001000. The formula is

fMHz = 1 / [6.28318 (LC)1/2]

where fMHz represents the resonant frequency in megahertz. Plugging the values in, we get

fMHz = 1 / [6.28318 (100 x 0.001000)1/2]
= 1 / (6.28318 x 0.11/2)
= 1 / (6.28318 x 0.316228)
= 1 / 0.986917
= 0.503 MHz

That's the equivalent of 503 kHz, so the correct choice is C.

7. Because the resistance makes no difference in the resonant frequency of an LC circuit, shorting out the resistor won't change it. The correct answer is C. The resistance does influence the "sharpness" of the resonant response, however. In a series RLC circuit, the "sharpness" increases as the resistance goes down; the "sharpest" possible resonant response occurs with the resistor shorted out. In a parallel RLC circuit, the opposite holds true; the "sharpness" increases as the resistance goes up, reaching its maximum when the resistor is removed altogether.
8. First, let's find the susceptance of a 100-H inductor at 318.31 kHz. Because we're given the inductance in microhenrys, it goes with megahertz in the formula

BL = -1 / (6.28318 f L)

where BL represents the inductive susceptance in siemens, f represents the frequency in megahertz, and L represents the inductance in microhenrys. In this case, f = 0.31831 MHz and L = 100 H, so

BL = -1 / (6.28318 x 0.31831 x 100)
= -0.00500 S

The resistance R equals 100 ohms, corresponding to a conductance G of 0.0100 S. We combine the conductance (as a real number) and the susceptance (as an imaginary number) to get

G + jBL = 0.0100 - j0.00500

The correct choice is B.

9. Let's start by calculating the susceptance of the 1000-pF capacitor at 3.1831 MHz. That capacitance equals 0.001000 F, a value that goes with megahertz in the capacitive-susceptance formula

BC = 6.28318 f C

where BC represents the capacitive susceptance in siemens, f represents the frequency in megahertz, and C represents the capacitance in microfarads. In this case, f = 3.1831 MHz and C = 0.001000 F, so

BC = 6.28318 x 3.1831 x 0.001000
= 0.0200 S

The resistance R equals 100 ohms, corresponding to a conductance G of 0.0100 S. We combine the conductance (as a real number) and the susceptance (as an imaginary number) to get

G + jBC = 0.0100 + j0.0200

The correct choice is A.

10. The component values in this example coincide exactly with those in the situation of Question 6, except this time, they're all connected in parallel rather than in series. In an RLC circuit, the resonant-frequency formulas for the series and parallel cases are identical. Therefore, the resonant frequency of this parallel combination is the same as the value that we got when we solved Question 6, that is, 0.503 MHz or 503 kHz. The correct choice is C.