Phasor Diagram for an
Inductor
„
„
„
The phasors are at
90o with respect to
each other
This represents the
phase difference
between the current
and voltage
Specifically, the
current lags behind
the voltage by 90o
Vector Addition of the Phasor
Diagram
„
„
„
Vector addition is
used to combine the
voltage phasors
ΔVL and ΔVC are in
opposite directions, so
they can be combined
Their resultant is
perpendicular to ΔVR
Total Voltage in RLC Circuits
„
From the vector diagram, ΔVmax can be
calculated
ΔVmax = ΔV + ( ΔVL − ΔVC )
2
R
2
= ( I max R ) + ( I max X L − I max X C )
2
ΔVmax = I max R + ( X L − X C )
2
2
2
Impedance
„
The current in an RLC circuit is
I max =
„
ΔVmax
R + ( X L − XC )
2
2
=
ΔVmax
Z
Z is called the impedance of the circuit
and it plays the role of resistance in the
2
2
circuit, where Z ≡ R + ( X L − X C )
„
„
Impedance has units of ohms
Also, ΔVmax = ImaxZ
Impedance Triangle
„
„
Since Imax is the
same for each
element, it can be
removed from each
term in the phasor
diagram
The result is an
impedance triangle
Impedance Triangle, cont.
„
The impedance triangle confirms that
Z ≡ R 2 + ( X L − XC )
„
2
The impedance triangle can also be used to
find the phase angle, φ
⎛ X L − XC ⎞
φ = tan ⎜
⎟
R
⎝
⎠
−1
„
„
The phase angle can be positive or negative
and determines the nature of the circuit
Also, cos φ = R
Z
Power in an AC Circuit
„
The average power delivered by the
generator is converted to internal energy in
the resistor
„
„
„
Pav = ½ Imax ΔVmax cos φ = IrmsΔVrms cos φ
cos φ is called the power factor of the circuit
We can also find the average power in terms
of R
„
Pav = I 2rmsR
*Note: “P” is used as the symbol for power here because the symbol used in the
text is not available on all computer platforms.
Power in an AC Circuit, cont.
„
„
The average power delivered by the source is
converted to internal energy in the resistor
No power losses are associated with pure
capacitors and pure inductors in an AC circuit
„
„
In a capacitor, during one-half of a cycle energy is stored
and during the other half the energy is returned to the
circuit and no power losses occur in the capacitor
In an inductor, the source does work against the back
emf of the inductor and energy is stored in the inductor,
but when the current begins to decrease in the circuit,
the energy is returned to the circuit
Power and Phase
„
„
The power delivered by an AC circuit
depends on the phase
Some applications include using
capacitors to shift the phase in heavy
motors so that excessively high
voltages are not needed
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
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
Power as a Function of
Frequency
„
Power can be expressed
as a function of frequency
in an RLC circuit
Pav =
„
( ΔVrms )
2
Rω 2
R ω + L (ω − ω
2
2
2
2
)
2 2
o
This shows that at
resonance, the average
power is a maximum
Active Figure 33.19
(SLIDESHOW MODE ONLY)
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 Pav
has half its maximum value
„
These points are called the half-power points
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
Typical Q values in
electronics range from 10
to 100
Transformers
„
„
An AC transformer
consists of two coils of
wire wound around a
core of soft iron
The side connected to
the input AC voltage
source is called the
primary and has N1
turns
Transformers, 2
„
„
The other side, called the secondary, is
connected to a resistor 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
„
„
Eddy current losses are minimized by using a
laminated core
Iron is used as the core material because it is a
soft ferromagnetic substance and reduces
hysteresis losses
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%
dΦB
ΔV1 = −N1
dt
In the primary,
The rate of change of the flux is the
same for both coils
Transformers, 4
„
„
„
„
The voltage across the secondary is
dΦB
ΔV2 = −N2
dt
The voltages are related by
N2
ΔV2 =
ΔV1
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
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 ⎠
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
„
In stereo terminology, this technique is
called impedance matching
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
„
A diode is a circuit element that conducts
current in one direction but not the other
Rectifier Circuit
„
„
The arrow on the diode (
) indicates the
direction of the current in the diode
Because of the diode, the alternating current in
the load resistor is reduced to the positive
portion of the cycle
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
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
„
„
It is determined by the RC time constant of
the circuit
This is represented by the dotted lines in
the previous graph
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
High-Pass Filter
„
„
The circuit shown is
one example of a
high-pass filter
A high-pass filter is
designed to
preferentially pass
signals of higher
frequency and block
lower frequency
signals
Active Figure 33.25
(SLIDESHOW MODE ONLY)
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
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
Quick Quiz 33.1
Consider the voltage phasor in the figure below,
shown at three instants of time. Choose the part of the
figure that represents the instant of time at which the
instantaneous value of the voltage has the largest
magnitude.
Quick Quiz 33.1
Answer: (a). The phasor in part (a) has the largest
projection onto the vertical axis.
Quick Quiz 33.2
For the voltage phasor in the figure below, choose the
part of the figure that represents the instant of time at
which the instantaneous value of the voltage has the
smallest magnitude.
Quick Quiz 33.2
Answer: (b). The phasor in part (b) has the smallestmagnitude projection onto the vertical axis.
Quick Quiz 33.3
Which of the following statements might be true for a
resistor connected to a sinusoidal AC source?
(a) av = 0 and iav = 0
(b) av = 0 and iav > 0
(c) av > 0 and iav = 0
(d) av > 0 and iav > 0
Quick Quiz 33.3
Answer: (c). The average power is proportional to the
rms current, which, as Figure 33.5 shows, is nonzero
even though the average current is zero. Condition (a)
is valid only for an open circuit, and conditions (b) and
(d) can never be true because iav = 0 for AC circuits.
Quick Quiz 33.4
Consider the AC circuit in the figure below. The
frequency of the AC source is adjusted while its
voltage amplitude is held constant. The lightbulb will
glow the brightest at
(a) high frequencies
(b) low frequencies
(c) The brightness will be the
same at all frequencies.
Quick Quiz 33.4
Answer: (b). For low frequencies, the reactance of the
inductor is small so that the current is large. Most of
the voltage from the source is across the bulb, so the
power delivered to it is large.
Quick Quiz 33.5
Consider the AC circuit in the figure below. The
frequency of the AC source is adjusted while its
voltage amplitude is held constant. The lightbulb will
glow the brightest at
(a) high frequencies
(b) low frequencies
(c) The brightness will be
same at all frequencies.
Quick Quiz 33.5
Answer: (a). For high frequencies, the reactance of the
capacitor is small so that the current is large. Most of
the voltage from the source is across the bulb, so the
power delivered to it is large.
Quick Quiz 33.6
Consider the AC circuit in this
figure. The frequency of the AC
source is adjusted while its
voltage amplitude is held
constant. The lightbulb will glow
the brightest at
(a) high frequencies
(b) low frequencies
(c) The brightness will be same
at all frequencies.
Quick Quiz 33.6
Answer: (b). For low frequencies, the reactance of the
capacitor is large so that very little current exists in the
capacitor branch. The reactance of the inductor is
small so that current exists in the inductor branch and
the lightbulb glows. As the frequency increases, the
inductive reactance increases and the capacitive
reactance decreases. At high frequencies, more
current exists in the capacitor branch than the inductor
branch and the lightbulb glows more dimly.
Quick Quiz 33.7
Label each part of the figure below as being XL > XC,
XL = XC, or XL < XC.
Quick Quiz 33.7
Answer: (a) XL < XC. (b) XL = XC. (c) XL > XC.
Quick Quiz 33.9
The impedance of a series RLC circuit at resonance is
(a) larger than R
(b) less than R
(c) equal to R
(d) impossible to determine
Quick Quiz 33.9
Answer: (c). At resonance, XL = XC. According to
Equation 33.25, this gives us Z = R.
Quick Quiz 33.10
An airport metal detector (see page 1003) is
essentially a resonant circuit. The portal you step
through is an inductor (a large loop of conducting wire)
within the circuit. The frequency of the circuit is tuned
to its resonance frequency when there is no metal in
the inductor. Any metal on your body increases the
effective inductance of the loop and changes the
current in it. If you want the detector to detect a small
metallic object, the circuit should have
(a) a high quality factor
(b) a low quality factor
Quick Quiz 33.10
Answer: (a). The higher the quality factor, the more
sensitive the detector. As you can see from Figure
33.19, when Q = ω0/Δω is high, a slight change in
the resonance frequency (as might happen when a
small piece of metal passes through the portal) causes
a large change in current that can be detected easily.
Quick Quiz 33.11a
Suppose you are designing a high-fidelity system
containing both large loudspeakers (woofers) and
small loudspeakers (tweeters). If you wish to deliver
low-frequency signals to a woofer, what device would
you place in series with it?
(a) an inductor
(b) a capacitor
(c) a resistor
Quick Quiz 33.11a
Answer: (a). The current in an inductive circuit
decreases with increasing frequency (see Eq. 33.9).
Thus, an inductor connected in series with a woofer
blocks high-frequency signals and passes lowfrequency signals.
Quick Quiz 33.11b
Remember, you are designing a high-fidelity system
containing both large loudspeakers (woofers) and
small loudspeakers (tweeters). If you wish to deliver
high-frequency signals to a tweeter, what device
would you place in series with it?
(a) an inductor
(b) a capacitor
(c) a resistor
Quick Quiz 33.11b
Answer: (b). The current in a capacitive circuit
increases with increasing frequency (see Eq. 33.17).
When a capacitor is connected in series with a tweeter,
the capacitor blocks low-frequency signals and passes
high-frequency signals.