alternating-current circuits chapter 22. section 22.2 ac circuit notation
TRANSCRIPT
Alternating-Current Circuits
Chapter 22
Section 22.2
AC Circuit Notation
PhasorsAC circuits can be
analyzed graphicallyAn arrow has a length
Vmax The arrow’s tail is at
the originThe arrow makes an
angle of θ with the horizontal
The angle varies with time according to θ = 2πƒt
Section 22.2
Phasors, cont.The rotating arrow represents the voltage in an AC
circuitThe arrow is called a phasorA phasor is not a vectorA phasor diagram provides a convenient way to
illustrate and think about the time dependence in an AC circuit
Section 22.2
Phasors, finalThe current in an AC
circuit can also be represented by a phasor
The two phasors always make the same angle with the horizontal axis as time passes
The current and voltage are in phaseFor a circuit with only
resistorsSection 22.2
AC Circuits with CapacitorsAssume an AC circuit
containing a single capacitor
The instantaneous charge is
q = C V
= C Vmax sin (2 πƒt)
The capacitor’s voltage and charge are in phase with each other
Section 22.3
Current in CapacitorsThe 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 functionsI = Imax sin (2πƒt + Φ) where Φ = π/2
Section 22.3
Capacitor Phasor DiagramThe current is out of
phase with the voltageThe angle π/2 is called
the phase angle,Φ, between V and I
For this circuit, the current and voltage are out of phase by 90o
Section 22.3
Current Value for a CapacitorThe peak value of the current is
The factor Xc is called the reactance of the capacitorUnits of reactance are Ohms Reactance and resistance are different because the
reactance of a capacitor depends on the frequencyIf the frequency is increased, the charge oscillated more
rapidly and Δt is smaller, giving a larger currentAt high frequencies, the peak current is larger and the
reactance is smaller
maxmax C
C
VI where X
X πƒC
1
2
Section 22.3
Power In A CapacitorFor an AC circuit with a
capacitor, P = VI = Vmax Imax sin (2πƒt) cos (2πƒt)
The average value of the power over many oscillations is 0Energy 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 InductorsAssume an AC circuit
containing a single inductor
The voltage drop is
V = L (ΔI / Δt)
= Vmax sin (2 πƒt)
The inductor’s voltage is proportional to the slope of the current-time relationship
Section 22.4
Current in InductorsThe instantaneous
current oscillates in time according to a cosine function
I = -Imax cos (2πƒt)
A plot of the current 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
Inductor Phasor DiagramThe current is out of
phase with the voltageFor this circuit, the
current and voltage are out of phase by -90o Remember, for a
capacitor, the phase difference was +90o
Section 22.4
Current Value for an InductorThe peak value of the current is
The factor XL is called the reactance of the inductor
Units of inductive reactance are Ohms As with the capacitor, inductive reactance depends
on the frequencyAs the frequency is increased, the inductive reactance
increases
maxmax L
L
VI where X πƒL
X 2
Section 22.4
Quiz!If the frequency in a
circuit with a Capacitor is halved.
The Reactance is?
A) DoubleB) The SameC) HalfD) ZeroE) Disproportionate
Section 22.4
Properties of AC Circuits
Power In An InductorFor an AC circuit with
an inductor, P = VI = -Vmax Imax sin (2πƒt) cos (2πƒt)
The average value of the power over many oscillations is 0Energy 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
LC Circuit
Most useful circuits contain multiple circuit elementsWill start with an LC circuit, containing just an
inductor and a capacitorNo AC generator is included, but some excess
charge is placed on the capacitor at t = 0Section 22.5
LC Circuit, cont.After t = 0, the charge moves from one capacitor
plate to the other and current passes through the inductor
Eventually, the charge on each capacitor plate falls to zero
The inductor again opposes change in the current, so the induced emf now acts to maintain the current at a nonzero value
This current continues to transport charge from one capacitor plate to the other, causing the capacitor’s charge and voltage to reverse sign
Eventually the charge on the capacitor returns to its original value
Section 22.5
LC Circuit, finalThe voltage and current in the circuit oscillate
between positive and negative valuesThe circuit behaves as a simple harmonic oscillatorThe charge is q = qmax cos (2πƒt)
The current is I = Imax sin (2πƒt)
Section 22.5
Quiz!If the frequency in a
circuit with an Inductor is doubled.
The Reactance is?
A) DoubleB) The SameC) HalfD) ZeroE) 42
Energy in an LC CircuitCapacitors and inductors
store energyA capacitor stores energy
in its electric field and depends on the charge
An inductor stores energy in its magnetic field and depends on the current
As the charge and current oscillate, the energies stored also oscillate
Section 22.5
Energy CalculationsFor the capacitor,
For the inductor,
The energy oscillates back and forth between the capacitor and its electric field and the inductor and its magnetic field
The total energy must remain constant
maxcap
qqPE cos πƒt
C C
2221 1
22 2
ind maxPE LI LI sin πƒt 2 2 21 12
2 2
Section 22.5
Energy, finalThe maximum energy in the capacitor must equal
the maximum energy in the inductor From energy considerations, the maximum value of
the current can be calculated
This shows how the amplitudes of the current and charge oscillations in the LC circuits are related
max maxI qLC
1
Section 22.5
Frequency Oscillations – LC CircuitIn an LC circuit, the instantaneous voltage across
the capacitor and inductor are always equalTherefore, |VC| = |I XC| = |VL| = |I XL|
Simplifying, XC = XL This assumed the current in the LC circuit is oscillating
and hence applies only at the oscillation frequencyThis frequency is the resonance frequency
resƒπ L C
1
2
Section 22.5
LRC CircuitsLet the circuit contain a
generator, resistor, inductor and capacitor in seriesLRC circuit
From Kirchhoff’s Loop Rule,
VAC = VL + VC + VR
But the voltages are not all in phase, so the phase angles must also be taken into account
Section 22.6
LRC Circuit – Phasor DiagramAll the elements are in
series, so the current is the same through each one
All the current phasors are in the same direction
Resistor: current and voltage are in phase
Capacitor and inductor: current and voltage are 90o out of phase, in opposite directions
Section 22.6
Resonance The VC and VR values
are the same at the resonance frequency
Only the resistor is left to “resist” the flow of the current
This cancellation between the voltages occurs only at the resonance frequency
The resonance frequency corresponds to the highest current
Section 22.6
Applications of ResonanceTuning a radio
Changes the value of the capacitance in the LCR circuit so the resonance frequency matches the frequency of the station you want to listen to
LCR circuits can be used to construct devices that are frequency dependent
Section 22.6
Real Inductors in AC Circuits
A typical inductor includes a nonzero resistanceDue to the wire itself
The inductor can be modeled as an ideal inductor in series with a resistor
The current can be calculated using phasors
Section 22.7
Real Inductor, cont.The elements are in series, so the current is the
same through both elementsVoltages are VR = I R and VL = I XL
The voltages must be added as phasorsThe phase differences must be included
The total voltage has an amplitude of
total R L LV V V I R X or
I R πƒL
2 2 2 2
22 2
Section 22.7
ImpedanceThe impedance, Z, is a measure of how
strongly a circuit “impedes” current in a circuitThe impedance is defined as Vtotal = I Z where
This is the impedance for an RL circuit onlyThe impedance for a circuit containing other
elements can also be calculated using phasorsThe angle between the current and the
impedance can also be calculated
Z R πƒL 22 2
Section 22.7
Impedance, LCR CircuitThe current phasor is
on the horizontal axisThe total voltage is
The impedance is
total L CV IR I X X
I R πƒLπƒC
2 22
2
2 12
2
Z R πƒLπƒC
2
2 12
2
Section 22.7
Resonance in an LCR CircuitThe current depends on the impedance,
Imax = Vmax /Z
Since the impedance depends on the frequency, the current amplitude also varies with frequency
For the maximum current, the impedance must be a minimum
The minimum impedance occurs when
πƒLπƒC
1
2 02
Section 22.7
Resonance, cont.Solving for the frequency gives
This is the same result as was found for the LC circuitThe maximum current occurs at the resonant
frequencyThis is the frequency at which the LCR circuit
responds most strongly to an applied AC circuit
ƒπ LC
1
2
Section 22.7
Behavior of Elements at Various Frequencies
Section 22.8
Elements and Frequencies, cont.Resistor
Resistors in an AC circuit behave very much like resistors in a DC circuit
The current is always in phase with the voltageCapacitor or inductor
Both are frequency dependentDue to the frequency dependence of the reactancesXC is largest at low frequencies, so the current
through a capacitor is smallest at low frequencies XL is largest at high frequencies, so the current
through an inductor is smallest at high frequencies
Section 22.8
RL Circuit ExampleWhen the input
frequency is very low, the reactance of the inductor is smallThe inductor acts as a
wire Voltage drop will be 0
At high frequencies, the inductor acts as an open circuitNo current is passedThe output voltage is
equal to the input voltage
This circuit acts as a high-pass filter
Section 22.8
RC Circuit ExampleWhen the input frequency
is very low, the reactance of the capacitor is largeThe current is very smallThe capacitor acts as an
open circuitThe output voltage is
equal to the input voltageAt high frequencies, the
capacitor acts as a short circuitThe inductor acts as a
wire The output voltage is 0
This circuit acts as a low-pass filter
Section 22.8
Quiz!
A) 2.0B) 0.5C) 1.0D) 0.1
Vac= 9VC1=10 uFC2= 2 uFF=2kHzHow many Amps?
Application of a Low-Pass FilterA low-pass filter is used in radios and MP3 playersA music signal often contains static
Static comes from unwanted high-frequency components in the music
These high frequencies can be filtered out by using a low-pass filter
Section 22.8
Frequency Limits, RL CircuitFor an RL circuit, the input frequency is compared
to the RL time constantThe time constant is τRL = L / RDefine a corresponding frequency as ƒRL = 1/ τRL = R / LThe high-frequency limit applies when the input frequency
is much greater than ƒRL A frequency higher than ~10 x ƒRL falls into the high-frequency
limitThe low-frequency limit applies when the input frequency is
much less than ƒRL A frequency lower than ~ƒRL / 10 falls into the low-frequency
limitSection 22.8
Frequency Limits, RC CircuitFor an RL circuit, the input frequency is
compared to the RL time constantThe time constant is τRL = R CDefine a corresponding frequency as ƒRC = 1/ τRC =
1 / RCThe high-frequency limit applies when the input
frequency is much greater than ƒRC A frequency higher than ~10 x ƒRC falls into the high-
frequency limitThe low-frequency limit applies when the input frequency
is much less than ƒRC A frequency lower than ~ƒRC / 10 falls into the low-
frequency limitSection 22.8
Filter Application – Stereo SpeakersMany stereo speakers actually contain two separate
speakersA tweeter is designed to perform well at high
frequenciesA woofer is designed to perform well at low
frequenciesThe AC signal passes through a crossover network
A combination of low-pass and high-pass filtersThe outputs of the filter are sent to the speaker
which is most efficient at that frequency
Section 22.8