35 AC Circuits

Recommended class days: 2

Background Information

AC circuits are an important application of electricity and magnetism, but no new physics is

required. This is an extension of the study of DC circuits in Chapter 31, and the same misunder-

standings about current and voltage will continue to haunt students unless those misunderstandings

were successfully dealt with in Chapter 31.

The title of this chapter, although conventional, is somewhat of a misnomer. Rather than

being about “household electricity,” as most students would surmise, the phasor analysis and the

majority of topics (e.g., resonance circuits) are more relevant to radiofrequency circuits.

Phasor analysis will be a new idea for nearly all students, and many find the idea rather difficult.

Students don’t readily distinguish between instantaneous values (v and i), peak values (V and I ) and

rms values (V and I ), and they’re often not sure which to use.

rms rms

Student Learning Objectives

To use a phasor analysis to analyze AC circuits.

To understand RC filter circuits.

To understand the series RLC circuit and resonance.

To calculate power loss in an AC circuit using the power factor.

Pedagogical Approach

All results in this chapter are derived from the geometry of a phasor diagram. Complex impedances

are not introduced. Students need to see careful explanations of phasors, the distinction between

peak and instantaneous values, and the idea of phase leads and lags.

AC circuits are much harder to demonstrate than DC circuits, but it’s well worth using a dual-

trace oscilloscope to show simultaneously both the voltage across and current through a circuit

element (using voltage through a series resistor as the current trace). In an RC circuit, for example,

you can show that changing the amplitude does not change the ratio V /V , but changing the fre-

R C

quency does. Frequency is usually the most important parameter for understanding AC circuits.

Using Class Time

DAY 1: This is a chapter easily covered in two days. Day 1 should concentrate on what a phasor is,

distinguish between peak and instantaneous values, and use phasors to understand resistor, capa-

citor, and RC filter circuits. The 90° phase difference between position and velocity in simple

harmonic motion is a good analogy for explaining the phase difference between voltage and

35-135-2 Instructor’s Guide

current in a capacitor. It’s important to emphasize that the reactance of a capacitor relates only the

peak voltage and current. Although reactance is much like resistance, resistance applies to both

peak and instantaneous values, whereas reactance applies only to peak value. Students easily lose

sight of the distinction.

An RC filter circuit is easily demonstrated with the dual-trace oscilloscope. Students find this

quite interesting if you relate it to the bass and treble knobs on their stereo. You could even have

students design a midrange filter that passes frequencies between 100 Hz and 1000 Hz by cascading

a high-pass filter and a low-pass filter, then build one and test it.

DAY 2: Day 2 can then focus mostly on inductor circuits and the RLC series circuit. Students like

the idea of resonance circuits because of their importance in telecommunications. A dramatic

demonstration is to watch the current of a high-Q circuit on the oscilloscope while sweeping the

frequency back and forth through resonance. Although the textbook derives the phase angle

and shows graphs, the fact that the current switches from leading to lagging as you go through

resonance is a fairly minor point. The resonance in the current amplitude is the most important

feature.

Power and the power factor are treated conventionally. The main point to get across is that no

net power is dissipated in an ideal capacitor or inductor. They alternately store and release energy as

the fields build and then collapse, but the average over a cycle is zero. All the power dissipation is

in the resistor, but the dissipation is less than a DC analysis would lead you to expect, due to the

fact that the current and voltage are usually not in phase. Hence the idea of the power factor.

Sample Reading Quiz Questions

1. The analysis of AC circuits uses a rotating vector called a .

2. In a capacitor, the peak current and peak voltage are related by the

a. capacitive resistance. c. capacitive impedance.

b. capacitive reactance. d. capacitive inductance.

3. In a series RLC circuit, what quantity is maximum at resonance?

a. The voltage. c. The impedance.

b. The current. d. The phase.Chapter 35: AC Circuits 35-3

Sample Exam Questions

These questions cover the material of Chapters 34–35.

1. Reference frame S′ moves along one of the coordinate axes of reference frame S.

Experimenters in frames S and S′ measure the electric and magnetic fields shown. Along

which axis does S′ move, and in which direction?

yy′

r

E

r

E′

r r

B B′

x x′

S S′

z z′

2

2. A 100 MHz radio wave with intensity 500 W/m is traveling in the positive y-direction.

a. Draw the magnetic field vector at the point shown.

b. What is the amplitude of the wave’s magnetic field?

y

x

r

E

z

3. A circuit with two elements, 1 and 2, has the phasor diagram shown in the figure. Each

element could be a resistor, a capacitor, or an inductor. Draw the circuit diagram.

0

I

V

1

V

2

4. Your supervisor asks you to build a low-pass filter with a crossover frequency of 500 rad/s.

You have two 50 Ω resistors, a 25 µF capacitor, and a 100 µF capacitor. Can you build the

circuit? If so, show the circuit diagram. If not, why not?

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