AC Circuits

15 April 2013

What we are going to cover

20.1 Resistors in an AC Circuit

21.2 Capacitors in an AC Circuit

21.3 Inductors in an AC Circuit

21.4 The RLC Series Circuit

21.5 Power in an AC Circuit

21.6 Resonance in a Series RLC Circuit

21.7 The Transformer

INTRODUCTION

• Every time we turn on a television set, a stereo system, or any of a multitude of other electric appliances, we call on alternating currents (AC) to provide the power to operate them.

• We begin our study of AC circuits by examining the characteristics of a circuit containing a source of emfand one other circuit element: a resistor, a capacitor, or an inductor.

• Then we examine what happens when these elements are connected in combination with each other.

• Our discussion is limited to simple series configurations of the three kinds of elements

INTRODUCTION

In an AC circuit, the charge flow reverses direction periodically.

In circuits that contain only resistance, the current reverses direction each time

the polarity of the generator reverses.

The output of an AC generator is sinusoidal and varies with time according to the equation:

21.1 RESISTORS IN AN AC CIRCUIT

An AC circuit consists of combinations of circuit elements and an AC generator or an AC source, which provides the alternating current.

Previously we have seen that The output of an AC generator is sinusoidal and varies with time according to the equation:

21.1 RESISTORS IN AN AC CIRCUIT• Let us consider a simple circuit consisting of a resistor and an

AC source (designated by the symbol . T .

• The current and the voltage across the resistor are shown in

Active Figure 21.2.

• To explain the concept of alternating current, we begin by discussing the current-versus-time curve in Active Figure 21.2.

• At point a on the curve, the current has a maximum value in one direction, arbitrarily called the positive direction.

• Between points a and b, the current is decreasing in magnitude but is still in the positive direction. At point b, the current is momentarily zero; it then begins to increase in the opposite (negative) direction between points b and c.

• At point c, the current has reached its maximum value in the negative direction.

• The current and voltage are in step with each other because they vary identically with time.

• Because the current and the voltage reach their maximum values at the same time, they are said to be in phase.

• Notice that the average value of the current over one cycle is zero. This is because the current is maintained in one direction (the positive direction) for the same amount of time and at the same magnitude as it is in the opposite direction (the negative direction).

However, the direction of the current has no effect on the behavior of the resistor in the circuit: the collisions between electrons and the fixed atoms of the resistor result in an increase in the resistor’s temperature regardless of the direction of the current

Power

• We can quantify this discussion by recalling that the rate at which electrical energy is dissipated in a resistor, the power is

• where i is the instantaneous current in the resistor.• Because the heating effect of a current is proportional to

the square of the current, it makes no difference whether the sign associated with the current is positive or negative.

• However, the heating effect produced by an alternating current with a maximum value of Imax is not the same as that produced by a direct current of the same value.

• The reason is that the alternating current has this maximum value for only an instant of time during a cycle.

RMSThe important quantity in an AC circuit is a special kind of average value of current, called the rms current—the direct current that dissipates the same amount of energy in a resistor that is dissipated by the actual alternating current.

To find the rms current, we first square the current, Then find its average value, and finally take the square root of this average value. Hence, the rms current is thesquare root of the average (mean) of the square of the current.

Therefore, the rms current I rms is related to the maximum value of the alternating current Imax by

• This equation says that an alternating current with a maximum value of 3 A produces the same heating effect in a resistor as a direct current of (3/2 ) A.

• We can therefore say that the average power dissipated in a resistor that carries alternating current I is

The mathematical part

• When we speak of measuring an AC voltage of 120 V from an electric outlet, we really mean an rms voltage of 120 V.

• A quick calculation using Equation 21.3 shows that such an AC voltage actually has a peak value of about 170 V. In this chapter we use rms values when discussing alternating currents and voltages.

• One reason is that AC ammeters and voltmeters are designed to read rmsvalues. Further, if we use rms values, many of the equations for alternating current will have the same form as those used in the study of direct-current (DC) circuits.

Consider the series circuit in Figure 21.1, consisting of a resistor connected to an AC generator. A resistor impedes the current in an AC circuit, just as it does in a DC circuit. Ohm’s law is therefore valid for an AC circuit, and we have

Problem

SolutionObtain the maximum voltage by comparison of the given expression for the output with the general expression:

Next, substitute into Equation 21.3 to find the rms voltage of the source:

Substitute this result into Ohm’s law to find the rms current:

21.2 CAPACITORS IN AN AC CIRCUIT

• In a purely resistive circuit, the current and voltage are always in step with each other. This isn’t the case when a capacitor is in the circuit.

• In Figure 21.5, when an alternating voltage is applied across a capacitor, the voltage reaches its maximum value one-quarter of a cycle after the current reaches its maximum value.

• We say that the voltage across a capacitor always lags the current by 90°.

Capacitive reactance

Note

Substitute the values of f and C

Solve Equation 21.6 for the current, and substitute Xc and the rms voltage to find the rms current:

Capacitors do not behave the same as resistors. Whereas resistors allow a flow of electrons through them directly proportional to the voltage drop, capacitors oppose changes in voltage by drawing or supplying current as they charge or discharge to the new voltage level. The flow of electrons “through” a capacitor is directly proportional to the rate of change of voltage across the capacitor. This opposition to voltage change is another form of reactance, but one that is precisely opposite to the kind exhibited by inductors.

21.3 INDUCTORS IN AN AC CIRCUIT• Now consider an AC circuit consisting only of an inductor

connected to the terminals of an AC source, as in Active Figure 21.6. (In any real circuit, there is some resistance in the wire forming the inductive coil, but we ignore this for now.)

• The changing current output of the generator produces a back emf that impedes the current in the circuit. The magnitude of this back emf is

• The effective resistance of the coil in an AC circuit is measured by a quantity called the inductive reactance,

• When f is in hertz and L is in henries, the unit of XL is the ohm.

The inductive reactance increases with increasing frequency and increasing inductance.

Inductors do not behave the same as resistors. Whereas resistors simply oppose the flow ofelectrons through them (by dropping a voltage directly proportional to the current), inductors oppose changes in current through them, by dropping a voltage directly proportional to the rate of change of current

Phase diagrams

Problem

Review of R, X, and ZResistance (R):

• This is essentially friction against the motion of electrons. It is present in all conductors to some extent (except superconductors!), most notably in resistors.

• When alternating current goes through a resistance, a voltage drop is produced that is in-phase with the current.

Reactance (X):

• It is essentially inertia against the motion of electrons. It is present anywhere electric or magnetic fields are developed in proportion to applied voltage or current, respectively; but most notably in capacitors and inductors.

• When alternating current goes through a pure reactance, a voltage drop is produced that is 900 out of phase with the current.

Impedance (Z)• This is a comprehensive expression of any and all forms of

opposition to electron flow, including both resistance and reactance.

• It is present in all circuits, and in all components. When alternating current goes through an impedance, a voltage drop is produced that is somewhere between 0o and 90o out of phase with the current.

Phasor Diagrams: Conventions

Phasor Diagrams

(a) A phasor diagramfor the RLC circuit

(b) Addition of the phasorsas vectors gives

(c) The reactance triangle that givesthe impedance relation

Summary

RESONANCE IN A SERIES RLC CIRCUIT

Questions

The diagram is given, then calculate the following :

(a) the impedance,

(b) The maximum current in the circuit,

(c) the phase angle, and

(d) the maximum voltages across the elements.

About average Power

Check RCL Summary as done in Example 4 : Voltages and power in a series RCL Circuit

page 720 on the 9th Edition

Chapter 23:Tutorial ExercisesProblem 1. A 63.0 µF capacitor is connected to a generator operating at a low frequency. The rms voltage of the generator is 4.00 V and is constant. A fuse in series with the capacitor has negligible resistance and will burn out when the rms current reaches 15.0 A. As the generator frequency is increased, at what frequency will the fuse burn out?

As the frequency f of the generator increases, the capacitive reactance XC of the capacitor

decreases, according to C

1

2X

f C

In terms of the rms current and voltage, Equation 23.1 gives the capacitive reactance as

rmsC

rms

VX

I . Substituting this relation into Equation (1) yields

Problem 17: A series RCL circuit includes a resistance of 275 Ω an inductive reactance of 648 Ω and a capacitive reactance of 415 Ω. The current in the circuit is 0.233 A. What is the voltage of the generator?

17. REASONING The voltage supplied by the generator can be found from Equation 23.6,

V I Zrms rms

. The value of Irms

is given in the problem statement, so we must obtain the

impedance of the circuit.

Z R X X 2 2 2275 3 60 10( – ) .L C

2 2( ) (648 – 415 )

The rms voltage of the generator is V I Zrms rms

A ) = 83.9 V ( . )( .0 233 3 60 102

The phase angle is given by tan = (XL – X

C)/R (Equation 23.8). When a series circuit

contains only a resistor and a capacitor, the inductive reactance XL is zero, and the phase

angle is negative, signifying that the current leads the voltage of the generator. The

impedance is given by Equation 23.7 with XL = 0 , or Z R X 2

C

2 . Since the phase

angle and the impedance Z are given, we can use these relations to find the resistance

R and XC.

Since the phase angle is negative, we can conclude that only a resistor and a capacitor are present. Using Equations 23.8, then, we have

C

Ctan or tan 75.0 3.73X

X R RR

According to Equation 23.7, the impedance is 2 2

C150Z R X

Substituting XC = 3.73R into thisexpression for Z gives

2 22 2 2

2

150 3.73 = 14.9 R or 150 14.9 R

15038.9

14.9

R R

R

XC = 3.73 (38.9 ) = 145 hence

Since, on the average, only the resistor consumes power, the average power dissipated in

the circuit is 2rms

P I R (Equation 20.15b), where Irms is the rms current and R is the

resistance. The current is related to the voltage V of the generator and the impedance Z of the

circuit by rms rms/I V Z (Equation 23.6). Thus, the average power can be written as

The impedance of the circuit is 22

L CZ R X X

2 2C

R X

Therefore, the expression for the average power becomes

2 2

2 2 2C

V R V RP

Z R X

C 6

1 12400

2 2 60.0 Hz 1.1 10 FX

f C

But

Hence

22

2 2 2 2C

120 V 2700 3.0 W

2700 2400

V RP

R X

Tutorials

Problem 52. When a resistor is connected by itself to an ac generator, the average power delivered to the resistor is 1.000 W. When a capacitor is added in series with the resistor, the power delivered is 0.500 W. When an inductor is added in series with the resistor (without the capacitor), the power delivered is 0.250 W. Determine the power delivered when both the capacitor and the inductor are added in series with the resistor.