Download Chapter 20 - Purdue Physics

Announcements
 Midterm Exam next Friday
 In class, ~1 hr.
 Closed book, one page of notes
 Bring a calculator (not phone, computer, iPad, etc.)
 Practice problems
 Available this weekend from course website
 Covers different types of problems
 Cannot cover everything, so still study notes, text, etc.
 No Class on Wednesday
 Independence Day Holiday
Chapter 22
Alternating-Current Circuits and Machines
Direct Current (DC) Circuit Summary
 Source of electrical energy is
generally a battery
 Current can vary with time but always
approaches a constant value after a
long time
 All circuits so far have been DC
circuits
AC Circuit Introduction
 AC stands for alternating current
 Current, voltage, etc. in circuit
vary with time
 There will be an associated
frequency and peak value
 Household electrical energy is
supplied by an AC source
 Standard frequency is 60 Hz
Generating AC Voltages
 Most sources of AC voltage employ a
generator based on magnetic
induction
 The changing flux through the coil
induces a voltage in the coil
 Generators convert the mechanical
energy of the rotating shaft into
electrical energy
 Conservation of energy still applies
Section 22.1
Generating AC Voltages
M
+Mmax
-Mmax
Section 22.1
Generating AC Voltages
Section 22.1
Values in AC Circuits
 Time-dependence requires referencing different time
scales for different values
 Instantaneous value: M
 Maximum value: Mmax

Occurs when
 Average value: Mave

In some cases, average is not useful (i.e. Mmin = -Mmax)
RMS Values
 RMS standard was adopted
 RMS stands for root mean squared
 For a time-dependent quantity, M
 The root-mean-square values are typically used to
specify the properties of an AC circuit
Section 22.2
AC Circuit Notation
Section 22.2
Resistors in AC Circuits
Section 22.2
Resistors in AC Circuits
 V = Vmax sin (2 π ƒ t)
 V is the instantaneous
potential difference
 Applying Ohm’s Law:
I =V
R
 Since the voltage varies
sinusoidally, so does the
current
 I = Imax sin (2 π ƒ t)
 Imax = Vmax / R
Section 22.2
Resistors in AC Circuits
 The instantaneous power is
 P = IV
 P = Vmax Imax sin2 (2πƒt)
 Since both I and V vary with
time, the power also varies
with time
Section 22.2
Resistors in AC Circuits
 The maximum power is then
 Pmax = Vmax Imax
 The average power is ½ the
maximum power
 Pavg = ½ (Vmax Imax ) = Vrms Irms
 Ohm’s Law can again be used to
express the power in different
ways
2
Vrms
2
Pave =
= Irms
R
R
Section 22.2
Capacitors in AC Circuits
Section 22.3
Capacitors in AC Circuits
 The instantaneous charge is
q=CV
= C Vmax sin (2 πƒt)
 The capacitor’s voltage and charge
are in phase with each other
 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
Capacitors in AC Circuits
 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
Inductors in AC Circuits
Section 22.4
Inductors in AC Circuits
 The voltage drop is
 V = L (ΔI / Δt)
 V = Vmax sin (2 πƒt)
 The inductor’s voltage is proportional to
the slope of the current-time relationship
 I = -Imax cos (2πƒt)
 Equivalently,
I = Imax sin (2πƒt + Φ) where Φ = -π/2
Section 22.4
Inductors in AC Circuits
 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
Current and Voltage in AC Circuits
 In capacitors and inductors, I and
Capacitor
V are out of phase
 I leads V in capacitors
 V leads I in inductors
 This out of phase relationship is
what causes no power to be
dissipated in these devices
 I and V are in phase in resistors,
so resistors dissipate power
Inductor
Reactance
 The peak value of the current in capacitors and
inductors is
 The factor X is called the reactance of the inductor
 Units of inductive reactance are Ohms
 Reactance depends on the frequency
 As the frequency is increased, the inductive reactance
increases
Section 22.4
Current Value for a Capacitor
 For capacitors,
 If the frequency is increased, the charge oscillated more
rapidly and Δt is smaller, giving a larger current
 At high frequencies, the peak current is larger and the
reactance is smaller
 For inductors,
 As the frequency is increased, the inductive reactance
increases
 At high frequencies, the peak current is larger and the
reactance is smaller
Section 22.3
Properties of AC Circuits
Section 22.4
LC Circuit
Section 22.5
LC Circuit, cont.
 The voltage and current in the
circuit oscillate between positive
and negative values
 The charge is q = qmax cos (2πƒt)
 The current is I = Imax sin (2πƒt)
 The circuit behaves as a simple
harmonic oscillator
 As the charge and current oscillate,
the energies stored also oscillate
 Energy stored in electric field of
capacitor depends on the charge
 Energy stored in magnetic field of
inductor depends on the current
Section 22.5
LC Circuit, cont.
 For the capacitor,
 For the inductor,
 The energy oscillates between the electric field of
the capacitor and the magnetic field of the inductor
 The total energy must remain constant
Section 22.5
LC Circuit, cont.
 From energy considerations, the maximum value of the
current can be calculated
Imax
1
=
qmax
LC
 Instantaneous voltage across the capacitor and inductor
are always equal in magnitude, but 180° out of phase
 There is a characteristic frequency at which the circuit will
oscillate, called the resonance frequency
Section 22.5