Chapter 4
Alternating Current Circuits
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Chapter 4:
4-1 AC Sources
4.2 Resistors in an AC
Circuit
4.3 Inductors in an AC
Circuit
4.4 Capacitors in an
AC Circuit
4.5 The RLC Series
Circuit
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4.6 Power in an AC
Circuit
4.7 Resonance in a
Series RLC Circuit
4.8 The Transformer and
Power Transmission
4.9 Rectifiers and Filters
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Objecties: The students should be able to:

Describe the sinusoidal variation in ac current and
voltage, and calculate their effective values.

Write and apply equations for calculating the
inductive and capacitive reactance for inductors
and capacitors in an ac circuit.

Describe, with diagrams and equations, the phase
relationships for circuits containing resistance,
capacitance, and inductance.
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
Write and apply equations for calculating the
impedance, the phase angle, the effective
current, the average power, and the resonant
frequency for a series ac circuit.

Describe the basic operation of a step up and a
step-down transformer.

Write and apply the transformer equation and
determine the efficiency of a transformer.
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4-7 Resonance in an AC Circuit

Resonance occurs at the frequency ωo where the
current has its maximum value



To achieve maximum current, the impedance must have a
minimum value
This occurs when XL = XC
Solving for the frequency gives
ωo  1

LC
The resonance frequency also corresponds to the
natural frequency of oscillation of an LC circuit
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Resonance, cont.



Resonance occurs at the
same frequency
regardless of the value of
R
As R decreases, the curve
becomes narrower and
taller
Theoretically, if R = 0 the
current would be infinite at
resonance
 Real circuits always have
some resistance
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Power as a Function of
Frequency

Power can be expressed as
a function of frequency in
an RLC circuit
av 

 Vrms 

2
Rω 2
R ω L ω ω
2
2
2
2
2
o

2
This shows that at
resonance, the average
power is a maximum
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Quality Factor


The sharpness of the resonance curve is
usually described by a dimensionless
parameter known as the quality factor, Q
Q = ωo / Δω = (ωoL) / R

Δω is the width of the curve, measured between
the two values of ω for which avg has half its
maximum value

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These points are called the half-power points
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Quality Factor, cont.

A high-Q circuit responds
only to a narrow range of
frequencies


Narrow peak
A low-Q circuit can detect a
much broader range of
frequencies
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4-8 Transformers


An AC transformer
consists of two coils of
wire wound around a
core of iron
The side connected to
the input AC voltage
source is called the
primary and has N1
turns
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Transformers, 2


The other side, called the secondary, is
connected to a resistor R and has N2 turns
The core is used to increase the magnetic
flux and to provide a medium for the flux to
pass from one coil to the other

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Eddy-current losses are minimized by using a
laminated core
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Transformers, 3

Assume an ideal transformer

One in which the energy losses in the windings
and the core are zero



Typical transformers have power efficiencies of
90% to 99%
d B
In the primary, v1  N1 dt
The rate of change of the flux is the same for
both coils
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Transformers, 4




The voltage across the secondary is
d B
v 2  N2
dt
The voltages are related by
N2
v 2 
v1
N1
When N2 > N1, the transformer is referred to as a
step-up transformer
When N2 < N1, the transformer is referred to as a
step-down transformer
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Transformers, 5

The power input into the primary equals the
power output at the secondary


I1ΔV1 = I2ΔV2
The equivalent resistance of the load
resistance when viewed from the primary is
2
 N1 
Req  
 RL
 N2 
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Transformers, final


A transformer may be used to match
resistances between the primary circuit and
the load
This way, maximum power transfer can be
achieved between a given power source and
the load resistance

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In stereo terminology, this technique is called
impedance matching
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4-9 Rectifier



The process of converting alternating current
to direct current is called rectification
A rectifier is the converting device
The most important element in a rectifier
circuit is the diode

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A diode is a circuit element that conducts current
in one direction but not the other
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Rectifier Circuit

The arrow on the diode (
) indicates the
direction of the current in the diode


The diode has low resistance to current flow in this
direction
Because of the diode, the alternating current in the
load resistor is reduced to the positive portion of the
cycle
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Half-Wave Rectifier

The solid line in the
graph is the result
through the resistor

It is called a half-wave
rectifier because
current is present in the
circuit during only half
of each cycle
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Half-Wave Rectifier,
Modification



A capacitor can be added to the circuit
The circuit is now a simple DC power supply
The time variation in the circuit is close to
zero


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It is determined by the RC time constant of the
circuit
This is represented by the dotted lines in the
previous graph
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4-10 Filter Circuit, Example


A filter circuit is one used to smooth out or
eliminate a time-varying signal
After rectification, a signal may still contain a
small AC component



This component is often called a ripple
By filtering, the ripple can be reduced
Filters can also be built to respond differently
to different frequencies
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High-Pass Filter

The circuit shown is
one example of a highpass filter

A high-pass filter is
designed to
preferentially pass
signals of higher
frequency and block
lower frequency signals
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High-Pass Filter, cont

At low frequencies, ΔVout
is much smaller than ΔVin


At low frequencies, the
capacitor has high
reactance and much of the
applied voltage appears
across the capacitor
At high frequencies, the
two voltages are equal

At high frequencies, the
capacitive reactance is
small and the voltage
appears across the resistor
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Low-Pass Filter



At low frequencies, the reactance and voltage
across the capacitor are high
As the frequency increases, the reactance and
voltage decrease
This is an example of a low-pass filter
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Summary:
1- If an AC circuit consists of a source and a resistor, the
current is in phase with the voltage. That is, the current
and voltage reach their maximum values at the same
time.
2- The rms current and rms voltage in an AC circuit:
where Imax and ∆Vmax are the maximum values.
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3- If an AC circuit consists of a source and an inductor, the
current lags behind the voltage by 90°. That is, the
voltage reaches its maximum value one quarter of a
period before the current reaches its maximum value.
4- If an AC circuit consists of a source and a capacitor, the
current leads the voltage by 90°. That is, the current
reaches its maximum value one quarter of a period
before the voltage reaches its maximum value.
 it is useful to define the inductive reactance XL and the
capacitive reactance XC as
where ω is the angular frequency of the AC
source. The SI unit of reactance is the ohm.
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5- The impedance Z of an RLC series AC circuit is
This expression illustrates that we cannot
simply add the resistance and
reactances in a circuit.
with the phase angle ϕ between the current
and voltage being
6- The sign of ϕ can be positive or negative, depending on
whether XL is greater or less than XC. The phase angle is zero
when XL =XC.
7- The average power delivered by the
source in an RLC circuit is
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8- The rms current in a series
RLC circuit is
9- The resonance frequency ω0
of the circuit is
The current in a series RLC circuit reaches its maximum value when
the frequency of the source equals ω0 —that is, when the “driving”
frequency matches the resonance Frequency.
10- Transformers allow for easy changes
in alternating voltage. Because energy
(and therefore power) are conserved,
we can write
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