Announcements Midterm Exam next Wednesday Exam starts at 6 PM, ~1 hr. Closed book, one page of notes Bring a calculator (not phone, computer, iPad, etc.) Practice problems Covers different types of problems Cannot cover everything, so still study notes, text, etc. No Class on Thursday or Friday Independence Day Holiday

Alternating-Current Circuits and Machines Chapter 22

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

Most sources of AC voltage employ a generator based on magnetic induction Generating AC Voltages Section 22.1

Generating AC Voltages Section M max - M max M

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: M max Occurs when Average value: M ave In some cases, average is not useful (i.e. M min = -M max )

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

Resistors in AC Circuits Section 22.2

Resistors in AC Circuits V = V max sin (2 π ƒ t) V is the instantaneous potential difference Applying Ohm’s Law: Since the voltage varies sinusoidally, so does the current I = I max sin (2 π ƒ t) I max = V max / R Section 22.2

Resistors in AC Circuits The instantaneous power is P = I V P = V max I max sin 2 (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 P max = V max I max The average power is ½ the maximum power P avg = ½ (V max I max ) = V rms I rms Ohm’s Law can again be used to express the power in different ways Section 22.2

Capacitors in AC Circuits Section 22.3

Capacitors in AC Circuits The instantaneous charge is q = C V = C V max sin (2 πƒt) The capacitor’s voltage and charge are in phase with each other The current is a cosine function I = I max cos (2πƒt) Equivalently, due to the relationship between sine and cosine functions I = I max sin (2πƒt + Φ) where Φ = π/2 Section 22.3

Capacitors in AC Circuits For an AC circuit with a capacitor, P = VI = V max I max 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 = V max sin (2 πƒt) The inductor’s voltage is proportional to the slope of the current-time relationship I = -I max cos (2πƒt) Equivalently, I = I max sin (2πƒt + Φ) where Φ = -π/2 Section 22.4

Inductors in AC Circuits For an AC circuit with an inductor, P = VI = -V max I max 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 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 Capacitor 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 Δ t 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

Section 22.4 Properties of AC Circuits

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 = q max cos (2πƒt) The current is I = I max 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 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

Mutual Inductance It is possible for the magnetic field of one coil to produce an induced current in a second coil The coils are connected indirectly through the magnetic flux The effect is called mutual inductance Section 21.4

Transformers Transformers make use of mutual inductance to 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 non-conducting 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 If the input coil has N in coils and the output coil has N out turns, the flux in the coils is related by The voltages are related by Section 22.9

Transformers, final The ratio of the turns can be greater than or less than one Therefore, the input voltage can be transformed to a different value Transformers cannot change DC voltages Change in current in DC circuit is zero (or very brief) 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 V out / V in is not affected by the presence of the magnetic material Section 22.9

Transformers and Power The output voltage of a transformer can be made much larger by arranging the number of coils By conservation of energy, the energy delivered through the input coil must either be stored in the transformer’s magnetic field or transferred to the output circuit Over many cycles, the stored energy is constant The power delivered to the input coil must equal the output power Section 22.9

Power, cont. P = V I if V out > V in, then I out < I in P in = P out only in an ideal transformer In real transformers, the coils always have a small electrical resistance which causes some power dissipation For a real transformer, the output power is always less than the input power (usually by only a small amount) Section 22.9

Applications of Transformers Transformers are used in the transmission of electric power over long distances Power dissipation in a electrical wire is P = V I DC voltage would waste too much energy in transmission (require large voltage and current) Transformers allow large AC voltage transmission with small current Many household appliances use transformers to convert the AC voltage at a wall socket to the smaller DC voltages needed in many devices Section 22.9