Ac circuits

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 emf
and 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.

RMS
The 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 the
square 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

• 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 rms
values. 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
Solution
Obtain 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°.

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 of
electrons 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

Review of R, X, and Z
Resistance (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
(a) A phasor diagram
for the RLC circuit
(b) Addition of the phasors
as vectors gives
(c) The reactance
triangle that gives
the impedance relation

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 Exercises
Problem 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
2
X
f C

In terms of the rms current and voltage, Equation 23.1 gives the capacitive reactance as
rms
C
rms
V
X
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 2
275 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 – XC)/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 3.73
X
X R R
R


     
According to Equation 23.7, the impedance is
2 2
C150Z R X   
Substituting XC = 3.73R into this
expression for Z gives
   
 
2 22 2 2
2
150 3.73 = 14.9 R or 150 14.9 R
150
38.9
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 2
rmsP 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 2
CR X 
Therefore, the expression for the
average power becomes
2 2
2 2 2
C
V R V R
P
Z R X
 

  C 6
1 1
2400
2 2 60.0 Hz 1.1 10 F
X
f C  
   

But
Hence    
   
22
2 2 2 2
C
120 V 2700
3.0 W
2700 2400
V R
P
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.

Ac circuits

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