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Chapter Review

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This chapter treats circuits with an alternating current (AC). These types of circuits are very important because AC circuits are used to provide the electricity that most of us use everyday. Included with this discussion is the behavior of resistors, capacitors and inductors in AC circuits.

24-1   Alternating Voltages and Currents

As discussed in a previous chapter, an AC generator supplies a voltage that switches (or alternates) in polarity. Typically, the voltage varies sinusoidally and can be represented by the expression

V - V_maxsinwt

where w is the angular frequency, w = 2pf, of the oscillating voltage and V_max is the amplitude of the varying voltage. When this voltage is connected in a circuit with resistance R, the result is a sinusoidally alternating current, that is, a current that alternates in direction according to

I = I_maxsinwt,

where, by Ohm's law, I_max = V_max/R. The voltage across the resistor in the circuit and the current vary together, reaching their maxima simultaneously, and equaling zero simultaneously. Because of this behavior, we say that the voltage across the resistor and the current in an AC circuit are in phase.

A convenient representation of alternating voltages is the use of phasors. A phasor is an arrow that rotates counterclockwise in an x-y coordinate system with an angular velocity w equal to the angular frequency of the alternating voltage it represents. The length of the arrow represents, and is proportional to, the amplitude of the alternating voltage. The projection of the phasor onto the y-axis gives the instantaneous value of the voltage at any given time. The current in an AC circuit can also be represented by a phasor, and because the voltage across the resistor and current are in phase, the phasor for the current always points in the same direction as the phasor for this voltage.

Since the voltage and current in an AC circuit alternate, it is customary to characterize quantities by an average value. However, the average values of the voltage and current are zero so instead we use the root mean square (rms) values. An rms value of a quantity is the square root of the average (mean) of the squared quantity. In the case of sinusoidally varying quantities, the rms value works out to be the maximum value divided by ,

V_rms = V_max/
I_rms = I_max/.

Using rms values, we can write equations for AC circuits that mirror equations we used for CD circuits. For example, an AC version of Ohm's law for a purely resistive circuit is

V_rms = I_rmsR.

Also, the average power consumed by the resistance R in an AC circuit is conviently written, in terms of rms values, by expressions that look just like the ones we used with CD circuits

P_av =I ²_rmsR = I_rmsV_rms = V ²_rms/R.

The instantaneous power used in the resistor is determined by the same expressions except we replace I_rms with I and V_rms with V. Also, any expression involving rms values can also be written for maximum values (V_max, I_max, P_max) simply by replacing the rms value of the quantity by its corresponding maximum value.

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Example 24.1   AC:   A simple circuit contains an AC generator with a maximum output of 150 V operating at a frequency of 60 Hz. If the resistance in the circuit is 35 W, what average power is dissipated through the resistor?

Picture the Problem   The picture shows an AC generator connected to a resistor.

Strategy   Of the three equivalent expressions for P_av we choose the one most convenient for the given information. Since we are given R and V_max we will use P_av = V ²_rms/R.

Solution

1. The rms voltage is:	Vrms = Vmax/ = (150 V)/ = 106.1 V
2. The average power then is:	Pav = V 2rms/R = (106.1 V)2/35 W = 320 W

Insight   Any of the three expressions could have been used to determine P_av, this one was most convenient because it involved the fewest intermediate steps.

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Practice Quiz

An alternating current is given by I = (0.25 A)sin[(377 rad/s)t]. What is the rms current? 0.25 A 0.18 A 377 A 0.12 A 0.50 A

An alternating current is given by I = (0.25 A)sin[(377 rad/s)t]. What is the frequency f? 377 Hz 0.25 Hz 60.0 Hz 0.18 Hz 2370 Hz

24-2 - 24-3   Capacitors in AC Circuits and RC Circuits

If you consider a circuit consisting of just an AC generator and a capacitor, with no resistance in it, the capacitor itself, because of charging and discharging, offers some opposition to the current being sent by the generator. This resistance-like opposition to the current due to the charging and discharging of the capacitor in an AC circuit is called the capacitive reactance X_C; it is given by

.

The capacitive reactance has the same SI unit as resistance does, the ohm W. Written in terms of the capacitive reactance, the rms current in the circuit is

.

Recall that, in a DC circuit, the capacitor takes time to change and discharge; similar behavior occurs in AC circuits as well. Because of this fact, there is a phase difference between the current and the voltage across the capacitor:

the voltage across a capacitor lags behind the current by 90^o.

Because of this lag, the phasor for the voltage across the capacitor is drawn 90^o behind the phasor for the current, with both phasors rotating counterclockwise, as shown below.

The key differences between the capacitive reactance and resistance are that (a) they result from very different types of physical processes, (b) the capacitive reactance depends on the frequency of the generator getting smaller as w increases, and (c) there is no net power consumption associated with X_C, as there is with R, because no net energy is used by the capacitor.

If we now consider an RC circuit run by an AC generator we must combine the considerations of subsection 24-1 with the present information. The voltage across the resistor V_R is in phase with the current, and is 90^o out of phase with the voltage across the capacitor. The phasors for these two voltages can be used to form a right triangle and the total voltage in the circuit can be determined using the Pythagorean theorem. For maximum values we have

.

This expression can also be written as . Making the definition

,

we can write this relation in a way that look's similar to Ohm's law,

V_max = I_maxZ.

The quantity Z is called the impedance of the circuit and represents the combined resistance-like effects of the actual resistance and the capacitance. The SI unit of impedance is the Ohm. The above expressions can also be written for rms values.

In general, the total voltage in an RC circuit will be out of phase with the current by an angle f between 0 and -90^o where the minus indicates that the voltage lags the current. Using trigonometry on the phasor diagram gives

;

these expressions can be used to find the magnitude of the angle f. Because energy is only consumed by the resistor the phase angle also comes into play in determining the average power consumed in the circuit. The result, equivalent to P_av = I ²_rmsR, is

P_av = I_rmsV_rmscosf;

The factor cos f is called the power factor.

Physlet Illustration: Capacitive Reactance
In this simulation, a 100 W resistor and a 1 mF capacitor are connected in series to a 10-Volt (maximum) AC source. The graph shows both the voltage and current  in the capacitor as functions of time. Vary the value of the source's angular frequency (500 rad/s < w < 5000 rad/s ) using the slider. How does this affect the current in the capacitor?
Hints: As you increase w to a very high frequency, does the current increase or decrease? Why? As you decrease w to a very low frequency, does the current increase or decrease? Why? Set a particular frequency. What is the capacitive reactance at this frequency? The impedance of the circuit? What, then, should be the maximum current? What is the phase relationship between the voltage and current in the capacitor? Does this relationship depend upon the frequency?

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Example 24.2   RC Circuits:   An AC circuit contains a 44.3 mF capacitor in series with a 50.1 W resistor. The circuit is powered by a 60.0 Hz generator with an rms output of 85.0 V. Find the rms current in the circuit and the average power consumed through the resistor.

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Picture the Problem   The picture shows a series RC circuit with an AC generator.

Strategy   We start with the basic expression for I_rms and find all the necessary quantities to use that expression. Then, we should also have enough information to determine P_av.

Solution

1. An expression for Irms is:
2. The capacitive reactance is:
3. The current then is:
4. The average power can be found as:	Pav = I 2rmsR = (1.089 A)2(50.1 W) = 59.4 W

Insight   The formula used for the rms current is just a rearrangement of the expression involving the maximum current and voltage with the maximum values replaced by rms values.

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Practice Quiz

An RC circuit contains a 25 W resistor and a 56 mF capacitor; it is run by a 60 Hz generator with a maximum emf of 120 V. What is the impedance of this circuit? 47 W 25 W 380 W 54 W 60 W

An RC circuit contains a 25 W resistor and a 56 mF capacitor; it is run by a 60 Hz AC generator with a maximum emf of 120 V. What is the rms current in this circuit? 1.6 A 2.2 A 3.4 A 1.8 A 2.5 A

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