LOGIN
AC Circuit Introduction
• AC stands for alternating current
– The power source is a device that produces an
electric potential that varies with time
– There will be a frequency and peak voltage
associated with the potential
• Household electrical energy is supplied by an
AC source
– Standard frequency is 60 Hz
• AC power has numerous advantages over
DC power
Introduction
DC vs. AC Sources
Introduction
Generating AC Voltages
• Most sources of AC voltage employ a generator
based on magnetic induction
• A shaft holds a coil with many loops of wire
• The coil is positioned between the poles of a
permanent magnet
• The magnetic flux through the coil varies with time
as the shaft turns
• This changing flux induces a voltage in the coil
• This induced voltage is the generator’s output
Section 22.1
Figure 22‐3b p751
Resistors in AC Circuits
• Assume a circuit
consisting of an AC
generator and a resistor
• The voltage across the
output of the AC source
varies with time
according to
V = Vmax sin (2 π ƒ t)
– V is the instantaneous
potential difference
– Vmax is the amplitude of
the AC voltage
Section 22.2
Resistors, cont.
• Applying Ohm’s Law:
I V
R
• Since the voltage
varies sinusoidally, so
does the current
Vmax
I
sin  2 ƒt  or
R
I  Imax sin  2 ƒt  where
Imax 
Vmax
R
Section 22.2
RMS Voltage
• To specify current and
voltage values when
they vary with time,
rms values were
adopted
– RMS stands for Root
Mean Square
• For the voltage
Vrms 
V 
2
avg
Vmax

2
Section 22.2
RMS Current
• The root-mean-square value can be
defined for any quantity that varies with
time
• For the current
Irms 
Imax
1 2
Imax 
 0.71 Imax
2
2


• The root-mean-square values of the
voltage and current are typically used to
specify the properties of an AC circuit
Section 22.2
QUIZ 1
In Japan, the rms voltage of household outlets is 100V.
What is the peak voltage?
a)
b)
c)
d)
e)
200V
141V
100V
50V
71V
Power
• The instantaneous
power is the product
of the instantaneous
voltage and
instantaneous current
P=IV
• Since both I and V
vary with time, the
power also varies with
time:
P = Vmax Imax sin2 (2πƒt)
PAVE = ½ PMAX
Section 22.2
Power, cont.
The instantaneous power varies between Vmax Imax and
0
The average power is ½ the maximum power
Pavg = ½ (Vmax Imax ) = Vrms Irms
This has the same mathematical
form as the power in
2
Vrms
2
Pave 
 Irms
R
a DC circuit
R
Ohm’s Law can again be used to express the power in
the same way
Section 22.2
AC Circuit Notation
• It is important to distinguish between
instantaneous and average values of voltage,
current and power
• The amplitudes of the AC variations and the rms
values are also important
Section 22.2
AC CIRCUIT WITH A SINGLE CAPACITOR
The instantaneous charge is q = C V = C Vmax sin (2 πƒ t)
The capacitor’s voltage and charge are in phase with each other
Figure 22‐10a p757
Current in Capacitors
• The instantaneous
current is the rate at
which charge flows
onto the capacitor
plates in a short time
interval
• The current is the
slope of the q-t plot
• A plot of the current
as a function of time
can be obtained from
these slopes
Section 22.3
Current in Capacitors, cont.
• The current is a cosine function
I = Imax cos (2πƒt)
• Equivalently, due to the relationship
between sine and cosine functions
• I = Imax sin (2πƒt + ϕ) where ϕ = π/2
Section 22.3
Peak Value of I for a Capacitor
•
The peak value of the current is
Imax 
•
•
•
•
Vmax
1
where XC 
XC
2  ƒC
The factor Xc is called the reactance of the capacitor
SI unit of reactance is the Ohm.
Reactance and resistance are different because the reactance of a capacitor
depends on the frequency
– If the frequency is increased, the charge oscillates more rapidly and ∆t is smaller,
giving a larger current
– At high frequencies, the peak current is larger and the reactance is smaller
Section 22.3
Power In A Capacitor
• For an AC circuit with a
capacitor, P = VI = Vmax
Imax sin (2πƒt) cos (2πƒt)
• The average value of the
power over many
oscillations is 0
– Energy is transferred from
the generator during part of
the cycle and from the
capacitor in other parts
• Energy is stored in the
capacitor as electric
potential energy and not
dissipated by the circuit
Section 22.3
AC Circuits with Inductors
• Assume an AC circuit
containing an AC
generator and a
single inductor
• The voltage drop is
V = L (∆I / ∆t)
= Vmax sin (2 πƒt)
• The inductor’s
voltage is proportional
to the slope of the
current-time
relationship
Section 22.4
Current in Inductors
• The instantaneous
current oscillates in
time according to a
cosine function
• I = -Imax cos (2πƒt)
• A plot of the currenttime relationship is
shown
Section 22.4
Current in Inductors, cont.
• The current equation can be rewritten as
I = Imax sin (2πƒt – π/2)
• Equivalently,
I = Imax sin (2πƒt + Φ) where Φ = -π/2
Section 22.4
Current Value for an Inductor
• The peak value of the current is
Imax
Vmax

where X L  2  ƒL
XL
• The factor XL is called the reactance of the
inductor
• SI unit of inductive reactance is Ohms
• As with the capacitor, inductive reactance
depends on the frequency
– As the frequency is increased, the inductive
reactance increases
Section 22.4
Power in an Inductor
• For an AC circuit with an
inductor, P = VI = -Vmax
Imax sin (2πƒt) cos (2πƒt)
• The average value of the
power over many
oscillations is 0
– Energy is transferred from
the generator during part of
the cycle and from the
inductor in other parts of
the cycle
• Energy is stored in the
inductor as magnetic
potential energy
Section 22.4
DEMO
ENERGY STORED IN AN INDUCTOR 6C‐07
Properties of AC Circuits
Section 22.4
Transformers
• Transformers are devices that can increase or decrease
the amplitude of an applied AC voltage
• A simple transformer consists of two solenoid coils with
the loops arranged so that all or most of the magnetic
field lines and flux generated by one coil pass through
the other coil
Section 22.9
Transformers, cont.
• The wires are covered with a nonconducting layer so that current cannot
flow directly from one coil to the other
• An AC current in one coil will induce an AC
voltage across the other coil
• An AC voltage source is typically attached
to one of the coils called the input coil
• The other coil is called the output coil
Transformers, Equations
• Faraday’s Law applies to both coils
 in
Vin 
t
and
Vout
 out

t
• If the input coil has Nin coils and the output
coil has Nout turns, the flux in the coils is
related by
 out
Nout

 in
Nin
• The voltages are related by
Vout
Nout

Vin
Nin
Section 22.9
Practical Transformers
• Most practical
transformers have
central regions filled
with a magnetic
material
• This produces a larger
flux, resulting in a larger
voltage at both the
input and output coils
• The ratio Vout / Vin is not
affected by the
presence of the
magnetic material
Section 22.9
Motors
• An AC voltage source can be use to power a motor
• The AC source is connected to a coil wound around
a horseshoe magnet
– Called the input coil
• The input coil induces a magnetic field that circulates
through the horseshoe magnet
Section 22.10
Motors, cont.
• A second coil is mounted between the poles of the
horseshoe magnet and attached to a rotating shaft
• The forces acting on the second coil produce a
torque on the coil
– This causes the shaft to rotate
• As the AC current in the input coil changes
direction, so do the forces
• The torques continue to produce a rotation that is
always in the same direction
• The oscillations of the AC current and field make
the shaft rotate
Section 22.10
QUIZ 2
Which of the following statements is correct ?
A| In A.C. circuit with a A.C. generator and an ideal capacitor the current lags the voltage.
B| In an A.C. circuit with a A.C. generator and an ideal inductor
the current leads the voltage.
C| In an circuit consists of an ideal inductor and an ideal capacitor.
The capacitor is initially charged. After the switch is thrown
energy is dissipated every cycle.
D| In an A.C. circuit consisting of an ideal inductor, an ideal capacitor and a series resistor. Initially the capacitor is charged After the switch is thrown energy is dissipated.
WARM UP QUIZ 2 Which of the following statements is correct? A.
I peak  2 I rms
V peak
1

Vrms
2
1
I rms

2
V peak  2Vrms
C.
I peak  2 I rms
V peak  2Vrms
D.
1
I rms

2
B.
I peak
I peak
V peak
1

Vrms
2