1. 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

2. Alternating-Current Circuits and Machines Chapter 22

3. 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

4. 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

5. Generating AC Voltages Section 22.1 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

6. Generating AC Voltages Section 22.1 +M max - M max M

7. Generating AC Voltages Section 22.1

8. 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 )

9. 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

10. AC Circuit Notation

11. Resistors in AC Circuits Section 22.2

12. 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

13. 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

14. 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

15. Capacitors in AC Circuits Section 22.3

16. 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

17. 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

18. Inductors in AC Circuits Section 22.4

19. 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

20. 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

21. 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

22. 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

23. 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

24. Section 22.4 Properties of AC Circuits

25. LC Circuit Section 22.5

26. 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

27. 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

28. 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